NB:- The River 'Thames' rhymes with 'gems'
But the River 'Thame' rhymes with 'name'

So here we name the River Thame
As one of the gems of the River Thames.
But one that starts a bit of a crisis
Because up from here some call Thames 'the Isis' JCE

In Oxford they give the main river the name of 'Isis'.  So, if you wish to be polite to those who hold it to be so, the river should be called 'Isis' from here to above Godstow.
See 'You are old, Father Thames' at St Johns Lock

1910: River Thame in Thames Valley Villages by Charles G Harper
Right Bank inflow of the River Thame which starts with a very tightly curved meander

River Thame Footbridge

1808: Turner -

River Thame Footbridge,Turner 1808
Union of the Thames and Isis ('Dorchester Mead, Oxfordshire) exhibited 1808

1811: The bridge had apparently been replaced -

Junction of the River Thames (aka Isis) and the River Thame, near Dorchester

River Thame Footbridge, from above -

River Thame Footbridge
On the River Thame from above its bridge under which it enters the Thames

The Thame is navigable for manually propelled boats at least to Dorchester, swans permitting. But recently at Dorchester the swans were difficult and there seems nowhere to stop that will allow access to the village.

1889: Jerome K Jerome -

Dorchester stands half a mile from the river.  It can be reached by paddling up the Thame, if you have a small boat.

River Thame
On the River Thame, 2004

1906: G.E.Mitton -

The Thame … Its arching trees and corners, and deep shady alleys, make it a delightful place for an idler. It runs close by the abbey church.

And suddenly we stumble on pure history - for fourteen centuries ago St Birinius arrived here in what was the territory of the fierce West Saxons and promptly converted King Cynegils of Wessex and baptised him and many of his people on this very spot on the Thame near Dorchester.

1890-1900: Dorchester Thame Bridge, Frederick John Hall -

Dorchester Thame Bridge, Frederick John Hall
Dorchester Thame Bridge, Frederick John Hall
© Oxfordshire County Council Photographic Archive; D211685a

1929: A Thames Survey -

Dorchester Bridge, eighteenth century, carries the London - Oxford road across the Thame about half a mile from its junction with the Thames; built of stone with stone balustrade and ramped approaches, good design and worthy of preservation. No increase in width required, as at present it tends to check speed of traffic on main road entering Dorchester.

1906: Mortimer Menpes, Dorchester Backwater -

Dorchester Backwater, Mortimer Menpes, 1906
Dorchester Backwater, Mortimer Menpes, 1906

Note the punting position.  This is the classic Thames punting position with the punt bows first, but standing in front of the till.  Since this is neither Oxford nor Cambridge style, and that elsewhere the punting clubs tend to specialise in racing punts, this position is going out of fashion. 
The other styles adopted today are:
the Oxford position (stern first);
the Cambridge position (bows first as above but standing on the till);
and of course the racing punt in which the punter stands centrally.
George Leslie, who has often been quoted in this book, used to “run” his punt, walking to the bows and then planting his pole and walking to the stern through the shove.  But he does mention the new-fangled Oxford style (1881). 
The “at rest” position shown has the pole held, not at the top as would be normally correct, but about a third of the way down.  This indicates that the water is shallow and is done to prevent the tip of the pole trailing on the bottom.
The Thame is much more overgrown now than shown here.

1610: Camden -

when a little beneath Oxford Isis and Cherwell have consociated their waters together within one channell, Isis, then entier of himselfe and with a swifter current, runneth Southward to find Tame whom so long he had sought for.
And gone he is not forward many miles, but behold Tame streaming out of Buckinghamshire meeteth with him, who is no sooner entred into this shire but hee giveth name to Tame, a mercate towne situate very pleasantly among the rivers. For Tame passeth hard by the Northside, and two riverets shedding themselves into it compasse the same, the one on the East, and the other on the West.
At the length Tame, by Haseley, where sometimes the name of Barentines flourished, as at Cholgrave, commeth to Dorchester, by Bede termed Civitas Dorcinia , by Leland Hydropolis, a name devised by his owne conceit, yet fit enough, considering that dour in the British tongue signifieth water.
That this towne was in old time inhabited by Romanes, their coined peeces of money oftentimes turned up doe imply, and our Chronicles record that it was for a long time much frequented by reason of a Bishops See, which Birinus the Apostle of the West-Saxons appointed to be there. For when he had baptised Cinigilse, a petie King of the West-Saxons, unto whom Oswald King of Northumberland was Godfather, both these Kings, as saith Bede, gave this Citie unto the same Bishop to make therein his Episcopal residencie. This Birinus, as we may read also in Bede, was wonderfully in those daies admired for a deepe conceived opinion of his holinesse, whereupon an ancient Poet who penned his life in verse wrote thus of him:

More worthy for to be extold than Hercules for might,
Or that great King of Macedon who Alexander hight:
For Hercules subdued his foes, and Alexander he
Wonne all the world by force of armes. But our Birinus, see,
Did vanquish both, nor conquerd he onely the world and so,
But in one fight subdu' d himselfe, and was subdued also.

After 460 yeeres, Remigus Bishop of this place, least the name of Bishop should loose credit in so small a city (a thing forbidden in the Canons), in the reigne of William the First translated his seat to Lincolne. At which time this city of Dorchester (as Malmesbury saith, who then flourished) De gestis pontificum Anglorum was but slender and of small resort, yet the majesty of the Churches was great, whether you respected either the old building or the new diligence and care emploied thereupon. Ever since it began by little and little to decay, and of late by turning London high way from hence, it hath decreased so as that of a citie it is scarse able now to maintaine the name of a towne, and all that it is able to doe is to shew in the fields adjoyning ruines onely and rubbish, as expresse tokens of what bignesse it hath beene.
A little beneath this towne Tame and Isis, meeting in one streame, become hand-fast (as it were) and joyned in Wedlocke, and as in waters, so in name they are coupled, as Jor and Dan in the Holy Land, Dor and Don, in France, whence come Jordan and Dordan. For ever after this the river by a compound word is called Tamisis, that is, Tamis. Hee seemeth first to have observed this who wrot the book entituled Eulogium Historiarum. Now as touching this marriage of Isis which you may read or leave unread at your pleasure:

Heere Zephyrus with fresh greene grasse
The Banks above doth spread,
Faire Flora with ay-living herbs
Adorneth Isis head.
Most lovely Grace selected forth
Sweet floures that never dy,
And gladsome Concord plats therof
Two gurlands skilfully.
With all God Hymenaeus lifts
His torches up in hie.
A Bride-chamber the Naiades
Beneath of rare device
And bed do reare, ywov' n with warp
Beset with stones of price,
Meane while, down Catechlanian hils
Tame, gliding, kindled had
The fire of love in hope of Ise,
Her husband wondrous glad.
Impatient now of all delay
She hastneth him to wed,
And thinks the daies be long untill
The meet in marriag-bed.
Untill, I say, ambitious she
May now before her love
Her owne name set: see whereunto
Ambition minds doth move!
And now by this shee leav' s the town
That known is by her name.
"All haile, fare well" redoubling to
The Norrises by the same.
Old Dorchester at length she sees
Which was to give presage,
And lucky Augury of this
Long wished marriage.
... then Tame and Isis both,
In love and name both one,
Hight Tamisis, more joys therein,
And hastning to be gone,
Ariseth up and leaping out
With hastfull hot desire,
Advanceth forth his streame, and seeks
The Ocean main his sire.

1653: Isaac Walton in The Compleat Angler quotes -

And, first, for the rivers of this nation: there be, as you may note out of Dr. Heylin's Geography and others, in number three hundred and twenty-five; but those of chiefest note he reckons and describes as followeth.
The chief is THAMISIS, compounded of two rivers, Thame and Isis; whereof the former, rising somewhat beyond Thame in Buckinghamshire, and the latter near Cirencester in Gloucestershire, meet together about Dorchester in Oxfordshire; the issue of which happy conjunction is Thamisis, or Thames; hence it flieth betwixt Berks, Buckinghamshire, Middlesex, Surrey, Kent and Essex: and so weddeth itself to the Kentish Medway, in the very jaws of the ocean. This glorious river feeleth the violence and benefit of the sea more than any river in Europe; ebbing and flowing, twice a day, more than sixty miles; about whose banks are so many fair towns and princely palaces, that a German poet thus truly spake:

We saw so many woods and princely bowers,
Sweet fields, brave palaces, and stately towers;
So many gardens drest with curious care,
That Thames with royal Tiber may compare.

1765: The Gentleman's Magazine -

To the Author of the remarks on the Account of Oxford, February 1765
... Having lived a long time in Oxford, I knew as well as you, that the main river in Oxford is called the Isis; and, as it now appears, better than you, that the true name of the river is the Tems, or Thames.
Your opinion that the name of the Thames is formed from Thame, and Isis, has indeed been universally received; and yet it is absolutely false.
Scholars have called the river Isis, but the country people, call it the Thames quite from its source
and in an old charter, granted to Abbott Aldhelm, particular mention is made of certain lands, upon the East part of the river,
cuius vocabulam Temis iuxta vadum quod appellatur Summerford
and as this Summerford is in Wiltshire, it is manifest that the river was then called Temis, or Tems, before its junction with the Thame.
The same thing appears in every charter and authentic history where this river is mentioned, particularly, in several charters granted to the Abbey of Malmsbury, and in some old deeds relating to Cricklade, both which places are also in Wiltshire.
All our historians, who mention the incursions of Aethelwold into Wiltshire in 905, or of Canute in 1016, tell us, that they passed over the Thames at Cricklade.
The Saxons called it Temese quite from its source, and from Temese our Tems, or Thames, is immediately derived.
The word seems to have been originally British, and as it is the name of several rivers in different parts of the island, particularly of the Thame, whose name the Isis has been supposed to borrow, the Tame in Staffordshire, the Teme which divides Shropshire and Herefordshire, the Tamar in Cornwall; and many others.
Mr Lloyd, the Welsh antiquary, affirms, that the Saxon Temese, was derived from their Taf, or Tavuys, a name common to many Welsh rivers, signifying a gentle stream. ...

It has to be said that the news that they do not live on the River Thames would be a surprise to people who live in Lechlade on Thames.  And indeed as you proceed up the Thames past the little bridge under which the Thame enters, you realise that the Thame makes little contribution, and what you are on is substantially the same river.  So what is this all about?  It is (in the archaic sense) a conceit - an imaginative fancy - brought about, one assumes, because the inhabitants of the ivory towers of Oxford preferred some classical allusion in the name of their river over the commonplace "Thames" with its barges and commerce.  See “Thames Bridge” above Oxford.

Daniel Defoe:

I shall sing you no songs here of the river in the first person of a water-nymph, a goddess, and I know not what, according to the humour of the ancient poets; I shall talk nothing of the marriage of old Isis, the male river, with the beautiful Thame, the female river (a whimsey as simple as the subject was empty).

However other authors are more weak minded: George Leslie -

It is not only the classic nature of the ground that makes this spot memorable;  here it is also that the Isis weds the Thame.
The junction of these two rivers is a very quiet sort of affair, the meek little Thame running into the Isis from beneath a humble tow-path bridge, reminding one of a wedding à la mode between aristocracy and plutocracy;  the Thame bringing little beside its name and title to its great rich bride the Isis, with her wealth of waters.
I explored the Thame for some distance up when I lived at Dorchester;  it is a quiet little river, with one or two mills on it of some beauty, but in general it runs along in a very ordinary way between rows of pollard willows.

1906: G.E.Mitton -

Tradition has it that the Thames proper does not begin until below Oxford, where it is formed by the junction of the Thame and the Isis. Tamese (Thames) means "smooth spreading water." Tam is the same root as occurs in Tamar, etc., and the "es" is the perpetually recurring word for water, e.g., Ouse, ooze, usquebagh. Isis is probably a back formation, from Tamesis.
In Drayton's Polyolbion, we have the pretty allegory of the wedding of Thame and Isis, from which union is born the sturdy Thames.

Now Fame had through this Isle divulged in every ear,
The long expected day of marriage to be near,
That Isis, Cotswold's heir, long woo'd was lastly won,
And instantly should wed with Thame, old Chiltern's son.

In Spenser's Faërie Queene the notion is carried one step further, and Thames, the son of Thame and Isis, is to wed with Medway, a farfetched conceit, for the rivers do not run into each other in any part of their course.

1909: The Stripling Thames by Fred Thacker -

The absurd old discussion about the name Isis I am almost ashamed to allude to. There is not a single title deed or folk tradition but employs the well loved name of Thames, and it only, in referring to the remoter stream. The deeds of Buscot Manor, to mention an unhackneyed instance, speak of the estate as "bounded by the Thames". If you are tracing your first footsteps by the Riverside [t]hereabouts, do not enquire for the Isis of the country people you meet. They do not know any such stream; and they are the true, the undeniable authority.

1889: Jerome K Jerome -

Dorchester is a delightfully peaceful old place, nestling in stillness and silence and drowsiness.  Dorchester, like Wallingford, was a city in ancient British times; it was then called Caer Doren, "the city on the water."  In more recent times the Romans formed a great camp here, the fortifications surrounding which now seem like low, even hills.  In Saxon days it was the capital of Wessex.  It is very old, and it was very strong and great once.  Now it sits aside from the stirring world, and nods and dreams.

Even more aside from the stirring world now it no longer has a main road to struggle with.

Dorchester Abbey, Mortimer Menpes, 1906
Dorchester Abbey, Mortimer Menpes, 1906

1932: England by Ronald Carton -

Roaming this country through we shall not do wisely to pass too quickly by the Abbey Church there, or hasten through the chain of fair towns and villages that cluster about the banks of the Thames all the way from Wallingford to Windsor – Goring, Pangbourne, Sonning, Henley, Marlow, Maidenhead, and the rest - with Reading, in piles of red brick, expressing commerce about halfway between the two, and shouldering its way aggressively down into Buckinghamshire. This is the region of river festivals, of low rounded hills and wooded slopes at the water’s edge, and villages a little way back from the stream and far enough from railways and main roads to escape the multitude.

Dorchester, The White Hart Hotel, Mortimer Menpes, 1906
Dorchester, The White Hart Hotel, Mortimer Menpes, 1906