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I would be surprised if any river could be navigated from source to sea, unless fed from a lake or glacier. The source of a river is usually defined as the most distant point following the water upstream from the mouth, and will almost always be very small indeed.
In the case of the Thames, at Cotswold Water Park it also becomes very fragmented, with lots of lakes of various depths, some of which have some form of public access.
Above about Lechlade, while there are pools that would take a small boat, the river isn't passable along its length. At Cricklade (Street View photo) the water doesn't look deep enough to kayak and even swimming would involve some walking the riverbed (I've swum at Marlow and Lechlade, but no higher; I might try near Cricklade this summer)
Hiring a self-drive boat (example, I know nothing about this company) you could tour from west London to Oxford. I did a bit of this about 30 years ago as a child. In London there are many sightseeing boats though getting to the estuary would be harder. Above Oxford there are rental options, but most aren't designed for continuous travel; I did have to watch out for some boats that were when swimming at Lechlade ,but these may have been privately owned.
In summary, if you want to explore the entire length of the Thames, you won't do it all by boat.
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The River Thames is navigable to "200.96 Limit of navigation". Not far downstream of that is Lechlade (on southern edge of the Cotswolds in Gloucestershire), where this Tripadvisor page shows boating.
You can generally rent boats by the hour, day or week, and in some places take organised river tours and cruises, along the navigable length of the Thames. Here is one example of many:
(no affiliation), but probably not continuously, although if you want to organise a trip from the source to the limit of navigation you may well be able to charter a small boat to take you, if you don't feel up to renting one yourself.
Edit: indeed there are ways to do it that can be arranged for you. Please see
River Thames: Hike, Bike & Kayak (Source to Sea)
Follow the iconic River Thames from the Cotswolds to Windsor
Pick up a paddle, lace up your boots and adjust your helmet for an action-packed adventure down Englandβs most iconic River β the Thames. Walk, cycle and canoe your way from the Cotswolds to Windsor, marvelling at quintessential British scenery and stopping off for plenty of pub lunches along the way. With a local leader at the helm, you can leave your worries in your wake as all the little details are taken care of. Explore charming villages, medieval towns and the fascinating city of Oxford, then in the evenings, rest your head in traditional inns filled with as much history as the destinations themselves.