I ran (and occasionally had to walk) the length of the Thames Path last year. That was over the course of four days - I started at the source, so stayed in Cirencester, then ran from source to Oxford on day one; Oxford to Henley on day two; Henley to Kingston on day three and Kingston to Greenwich on day four.
That was over the course of the four-day weekend we had for the jubilee, so at that time of year (early June) there wasn't a problem with flooding aside from a small stretch near the source. I believe that's possibly not the case now. Not sure what time of year you're planning on but clearly summer is probably going to be a bit drier.
The Thames Path is generally well marked and there shouldn't be much of a problem following it, though an OS map can be very handy at times. The source to Oxford is the trickiest bit in terms of wayfinding.
I had hugely valuable support along the way from my very understanding wife, who drove to meet me at various rendezvous points, meaning I didn't have to take all my food, drink and clothes in a bag on my back.
Happy to provide further info if you want to drop me a message.
Would echo the above comments - either try to get public transport or arrive with plenty of extra time to spare if you need to park your car. There's usually congestion when the rugby field is available for parking but if it's not then clearly things will be worse.
I'm in the fortunate position of being able to walk to the start line but, knowing the area well, I'm struggling to see how all the cars are going to be parked in reasonably close proximity to the start/race HQ.