Cookham Lock Cut: Lock cut on the Left bank
Hedsor Water is the weir stream straight on upstream (or coming downstream turn left after lock)
1834: Tombleson -

Hedsor, 1834, Tombleson

1859: The Thames, Mr & Mrs Hall

The woods of Hedsor — the seat of Lord Boston — companion us for a long way, and for some miles we keep in sight a remarkable structure which crowns the summit of a hill, — we learn that it is nothing more than a summer-house, placed there for the sake of the many views it commands; but it looks like the huge ghost of some mighty edifice which man has deserted.

1881: George Leslie, "Our River" -

Lord Boston's eel-bucks in the mainstream at Hedsor prevent boats passing up the water to the right, though I believe there is a perfect right of way, so long as no landing is attempted.  Formerly the barges went round this way, and when the lock was made, the then Lord Boston brought an action against the Thames Conservancy for loss of toll, which were claimed at Hedsor.
An ineffectual attempt was once made to turn me away, whilst sketching in my punt just below the bucks.  It is quite absurd to pretend that the water here is in any sense private.  There are numerous attempts made all up the river to shut up back-waters that run through private property;  sometimes bars are put across, and worst of all, wires, which are extremely dangerous in the dark;  if any fatal accident should occur, I believe the proprietors of the wire would be morally guilty of manslaughter.  The selfishness of those who have the use and pleasure of all other parts of the river in thus endeavouring to prevent one or two boats from occasionally coming past their property, seems to me very inexcusable.

1889: A Pictorial History of the Thames, A S Krausse -

[Coming downstream] on turning to the left out of Cookham Lock, it will be seen that the main river is closed to navigation by some eel-bucks, which extend right across the river, thus preventing boats coming up as effectually as does the upper weir ... those coming down. In this way about ¾ mile of the main river Thames is shut out from the public in the face of the provisions of the Thames Preservation Act, 1884 ... It may seem curious that such a state of things should be permitted, and it is therefore desirable that the means by which this has been effected should be explained.
The portion of the river in question surrounds the property of Lord Boston, and is known as Hedsor, the house standing on the summit of the hill commanding extensive views. The property has been in the Boston family since the year 1682, and in olden times, before the present lock-cut was made, the then Lord Boston used to levy a toll on all horses towing barges on the river past his ground, and this toll was let by him at a yearly rental of £73-10s.
In 1830, the then existing River Commissioners decided to make a lock-cutting, and as this caused a diversion of the barge traffic, Lord Boston claimed compensation amounting to £2,112 10s. in all. The case came into the law courts, where Lord Boston's claim was reduced considerably, and he was awarded £1,000 and £200 costs.
In September, 1837, the upper weir was erected with a view to improving navigation and keeping back a sufficient head of water, and Lord Boston again made a demand for compensation on the ground that he could not let his private wharves, as barges were unable to approach that portion of the river owing to the weir. After considerable negotiation, the matter was settled by the Thames Commissioners agreeing to pay Lord Boston £75 and undertaking to make a lock in the upper weir for his use.
The upper weir was rebuilt in 1869, and the lock done away with, but the eel bucks erected by Lord Boston at the lower end, together with a small weir alongside are untouched; the river between these two is entirely cut off from the public and given up to the private use of Lord Boston and his nominees.
These are the bare facts of the case in question, and further details will be found in the evidence given before the Select Committee of 1884. The fact remains that, at the present moment, upwards of half a mile of the main river Thames, situated amidst the most beautiful scenery on its banks, is closed to the public and delivered over to the solitary enjoyment of a single individual.
It is only fair to state that Lord Boston is usually ready to give permission to fish in his private water on application being made, but this does not affect the fact that all access to the Hedsor River is denied.

1889: A Pictorial History of the Thames, A S Krausse, prints this print of Hedsor Water in 1820 -

Hedsor Water, 1820

2003: And the story continues  (Newspaper report) -

The widow of the late Lonrho boss Tiny Rowland on 20th December 2003 lost a long legal action to protect the privacy she had been led to expect at her magnificent home by the Thames.
Navigation authorities had agreed that for more than a century, Hedsor Water near Cookham, Berkshire, was the only stretch of the river not subject to any public rights of navigation. But the Environment Agency, which now looks after the Thames, told Josie Rowland in 2001 that she had no such rights to privacy and to remove signs put up to exclude the public.
In June that year she began a legal action which ended yesterday in the court of appeal, where a judge said it was "unjust" that the courts could not help her. Lord Justice May said the Environment Agency and its predecessors had acted for more than a century in a way that gave Mrs Rowland the right to expect that Hedsor Water was private.
But because the navigation authorities had no power to overturn a legal public right of navigation which applies to the whole of the Thames, English law could not help Mrs Rowland. The judge said: "This is unjust and illustrates a defect in the law. In my view, the just outcome of these proceedings is that Hedsor Water should remain private. English law cannot at present achieve this."
Lord Justice Peter Gibson, giving the background to the case, said the couple paid £351,997 for Hedsor Wharf in 1968 and Mrs Rowland had continued to live there since the death of her husband in 1998. Included in the sale was Hedsor Water, which had been bypassed by a channel cut in 1830 to ease navigation on the Thames. Although public rights of navigation have existed on the river for a long time, it was treated as private by navigation authorities. A high court judge ruled that the public right could only be changed by legislation.

On appeal the case was again lost, however the Environment Agency were told not to actively promote navigation on Hedsor Water -

... the declaration granted to the defendant would be amended to provide that it must take into account, in exercising its statutory functions, the common assumption previously held by both parties that the stretch was private.  It was sufficient for the defendant [the Environment Agency] to act in a manner sensitive to the claimant's expectation by not actively promoting public use of Hedsor Water.

The House of Lords has dismissed a petition by the claimant for leave to appeal.
The Appeal is set out here
So I interpret that as:
Can you use Hedsor Water? Yes - but for legal reasons don't expect a clear reply if you ask the Environment Agency this question. The very helpful Cookham lock keeper is well aware of the legal situation and will carefully answer questions as best he can.
Should you use Hedsor Water?
Well it doesn't go anywhere much and we should all be sensitive to the secluded nature of this lovely stretch.  It is not in any case in my opinion a suitable place for powered vessels of any description.   Before the footbridge was rebuilt there were stones and piles below the lower footbridge. (These may well have been the remains of the eel bucks.) These have now I think been cleared away
Landing is of course a matter for the land owner and it does, on the face of it, seem unlikely that permission would be granted along here. Anchoring and remaining stationary for a reasonable period is part of the public right of navigation.
[ I am no lawyer and the above merely represents my personal effort to summarise the situation; it should not be regarded as in any way authoritative. ]

2003: In the Court of Appeal -

The Lower Weir has now virtually disappeared from sight through disrepair with only remains of it appearing in the water. It would appear to be a hazard but not an insuperable obstacle to those wishing to enter Hedsor Water with boats of a shallow draught, at least when the water is high.

When I punted up Hedsor Water in August 2005 there was still in the middle of the river the old notice -

Hedsor notice, photographed on 2nd August 2005

However by October 2005 a  "Danger" sign had been placed over the old sign.  It is unusual to mark the downstream end of a weir stream with a danger sign - however there is danger here - danger at least of grounding on the shallow bank in the centre of the river just above the sign.

Hedsor notice, photographed on 8th October 2005

Before 2014: Hedsor lower Footbridge -

Hedsor Lower Footbridge before 2014

2014: Under the central arch is a bird sitting on a timber pile at the head of a large and awkward pile of stones with around it a much higher current than found elsewhere in this reach.  Beware when negotiating this bridge. There is a line of piles at or just below water level across the stream immediately downstream of the bridge.

2017: The danger sign has gone but the post remains.

Hedsor Lower footbridge 2017

Hedsor Lower Footbridge

Straight ahead is a footbridge -
This has now had what appears to be a thorough rebuild. Beware of strong currents here. Not suitable for powered boats.

Hedsor Notice Autumn 2017

The notice has now changed:

Hedsor Notice Autumn 2017

1875, Eel Traps at Hedsor Weir, Henry Taunt. The weir referred to is Hedsor lower weir which no longer exists as such. It was almost certainly where the lower footbridge is now.

Eel Traps at Hedsor Weir, Henry Taunt, 1875
© Oxfordshire County Council Photographic Archive; HT01833

1881: George Leslie, "Our River", Hedsor Eel Bucks -

The Eel Bucks at Hedsor, George Leslie, 1881

1775:  Corporation of London Survey: "Hedsor bucks a very dangerous place;  three barges sunk within the last year".
1794:  "One of the most difficult and dangerous places on the Thames between Reading and Boulter's Lock is from Cookham Ferry to Clifden Wall".
1826:  Jos Gibbins of Abingdon, bargemaster -

Begs to state that the Mary, one of his barges, was coming to London laden with Bath Stone for Westminster, and in the night of the 9th of August, the water being very low and the night dark;  at Hedsor, she got aground and swung across the channel, which caused her to be broken in two.  Loss about forty pounds.

Round the bend, on the Right bank, is the large house referred to above.  I did not photograph it.

Hedsor Upper Footbridge

A second footbridge (with a tree down beyond it; this was very promptly dealt with)

Upper Hedsor Footbridge

Hedsor (Odney) Weir

Hedsor Weir from below.

Over the weir fall on the right can be seen the quirky blue of Cookham Bridge, and above the next two falls can be seen The Ferry Inn.

1883: Odney Weir, Henry Taunt -

Odney Weir, Henry Taunt, 1883
© Oxfordshire County Council Photographic Archive; HT3594

Coming back down Hedsor Water -

Coming back down Hedsor Water

Looking towards Cliveden from Hedsor Water -

1808: Turner painted a picture entitled 'Cliveden on the Thames'. However I reckon it was taken from more or less the position above, looking down Hedsor Water facing East South East. The trees make it difficult to tell - maybe another hundred yards downstream. But anyway I think this is Hedsor and Cliveden House was a burnt out ruin at this time -

Hedsor & Cliveden from "The Genius of the Thames" by Thomas Love Peacock:

Till rise romantic Hedsor's hills,
And Cliefden's groves, and springs, and rills

Hedsor House

1885: The Royal River -

In the distance, to the left of Cliefden [going downstream], and seemingly forming with it one demesne, lies Hedsor Park, the seat of Lord Boston, with its imitation castle, which would be improved, pictorially speaking, by a judiciously adminstered dose of dynamite. Hedsor overlooks the old course of the river.

[I take no responsibility for words written in 1885 - I merely quote.]

Hedsor Lodge, looking towards Maidenhead. June 1, 1793.
J. Farington R.A. delt. J.C. Stadler sculpt.
(Published) by J. & J. Boydell, Shakespeare Gally. Pall Mall & (No. 90) Cheapside (London)

In 2005 I wrote "Hedsor Park is nowhere visible from the Thames now, as far as I know." I now know better. Look very carefully in the centre of this 2006 photo and you can just glimpse it. It is taken from above Hedsor upper footbridge.

Hedsor House just glimpsed through trees from Hedsor Water

1881: George Leslie, "Our River" -

At the end of the Cliveden reach there are three back-waters, one with a very sharp stream up to the mill on the left-hand side, another of a private character behind the island of Formosa, and a very beautiful one just beside the lock which leads to Odney Weir;  this last is well worth exploring. 
Camping parties are generally to be seen on the land about the lock.  I like these little camps, but I wish the picnic people would take the trouble to clear up and burn the bits of paper and straw bottle-covers, which are so frequently seen vulgarizing the prettiest parts of the river banks.

Oh yes - the picnic people - bits of paper and straw bottle covers - those were the days when we were just practising - before we really learnt how to make litter!