You have got us thinking, Fallingstar.
I get the impression you would rather stay in London, for important family reasons.
Posters have commented that it can get too hot up north as well.
By the time you arranged a move, the weather would be cooler.
From what I read, you need 3 things.
A cold room.
A solution to the overheating conservatory.
A way to make your back yard comfortable.
Air cons may be sold out just now, but they will come back into stock. Research which one is best for you. Some Posters have helped with this upthread.
Check out the suggestions, again upthread, about cooling your conservatory.
How do people in hot countries keep their gardens/back yards cool? Research online.
Worried about not being able to go outside to exercise? Just remember that we all managed the 12 week lockdown.
Good luck, and let us know how you get on.