This refers to South London only. You should probably be looking to the north, Epping area. Especially if you want to compete.
There is a livery and dressage yard at Ham House Stables and another at Petersham Farm. I havent up to date info but Petersham Farm used to be adult orientated. Neither yard has an indoor school. The dressage at both is classical, Spanish style. I dont think they compete.
The only indoor school I have ridden in recently was at Chessington, and that was pre Covid. There is a riding school and an attached livery yard with competition horses, and competition instruction but you would need to enquire.
Kingston riding school has moved but it used to have a specialist dressage instructor and liveries who owned dressage horses.
My own less grand dressage lessons were at Stag Lodge on the A3 on a very willing pony. I heard that Stag Lodge now offer livery but I havent been there to enquire. And the owner evented in those days, the dressage being a component of eventing.
In fact I havent had a proper riding lessons for years as I love hacking and have hacked two long term shares, riding on my own which means I dont have riding companions or hear what is going on.
As for driving out from central London, it isnt easy, which is why most people live in the comuter suburbs. Or use public transport to reach their stables.
I have ridden in Hyde Park but I dont know if the yards there take livery horses and they dont have the sort of schools and turn out that you are looking for.