Have a look at Barnes Eagles. https://www.barneseagles.co.uk/
 Barn Elms Sports Trust, 
Barn Elms Playing Fields, 
Queen Elizabeth Walk
Barnes, London SW13 9SA. 

Free parking with a café at the clubhouse.
No dogs are allowed anywhere on the Barn Elms site.

Hi, just seen this. I'd recommend Clapham Rangers. They are a great club and a well run, in my experience. My son has been playing for them for 2 years now and attended their development  training before that. They are based around Clapham common though, so I'm not sure if that is outside your target area.

https://claphamrangersfc.com/

My son enjoys playing football and has played with a local club for a few years, but feels that he’s not really progressing anymore and would like to find a new club that will challenge him.

Does anyone have any recommendations? Ideally somewhere in the Wimbledon/Wandsworth area, but happy to travel a bit further for the right club.

Thanks