There's a place in Nottingham (I know, not M25, but i'll get to that) called Kitty Cafe where a pal of mine adopted a couple of cats from. I think the premise is that you can go there for an afternoon and interact with them to get a better measure of them than just a quick look through a grille in a rehoming shelter.
From what I understand these kinds of places are becoming quite popular and a lot more are popping up. Might be worth looking for something similar closer to you? Also would be a great treat for the kids.
https://kittycafe.co.uk/