You should be fine.
This thread is a good place to start:
Next step is to bin your Drone Assist app and use Drone Scene instead:
Why? It will let you switch off Class D Airspace, which covers much of London including most of Tooting, but is not a restriction on drone flights.
Then you’ll be left with something like this:
The big red circle is the London heliport Flight Restriction Zone. You can’t fly there without the heliport’s permission.
The red blob just at the top of the image at Nine Elms is the edge of the Hyde Park Restricted Area. You can’t fly there without CAA permission.
The remaining red area with angular edges is a restriction on helicopters - it does not affect you and you can fly in this area so long as you follow the normal drone regulations.
From Drone Scene, you can use the tools to deduce that you are probably within the boundary of Wandsworth Council, and click on a link that will show if the famous Simon Hawkins has discovered any drone restrictions that Wandsworth may impose (it shows he has not asked them yet).
Wandsworth’s website will tell you they don’t allow drones in their parks or streets, but their byelaws don’t seem to prevent it.