Tokyo89 welcome, it's great when new posters join in! i hope you're enjoying your baby - I remember my own style and taste went a bit nuts for several years afterwards,, I didn't lose my love of clothes but I made some very odd choices.
Hope all is well for you Auld?
OK London charity shops - I shouldn't really be joining in because I've not been for so long so just bear in mind this is Old and possibly Dodgy Information: there was a good Farah shop on Elgin St, can't remember which district that is but fairly central. And yes the Pimlico branch is chaotic but has quite a lot of vintage, and there are several other charity shops in the neighbourhood.
Marylebone has several fairly close together along or just off the high street. (It's also near The Wallace Collection where Vivienne Westwood was said to do a lot of research but I never saw her.)
Wimbledon Village - a bus ride away from any Wimbledon tube stations, several in a very compact area, not exactly bargainous and a bit boardroom / charity dinner for my taste but that was pre- pandemic and things my have loosened up. And if you want to look at some new labels there's also a Matches, but it was closed by the time I found it so I can't report back on how welcoming they are towards browsers.
Brixton - I wouldn't make a special trip but if you're in or close to the neighbourhood they have a big Barnardo's that has a really extensive range from cheap to high end. There's a Farrah too. If I remember correctly (but I may not) it's only a couple of stops from Pimlico so it might be possible to combine them.
Kensington, yes quite a few good ones although very spaced out as I remember so pick a nice day and be prepared to walk.
As far as specific shops go, Trinity Hospice has already been mentioned - not cheap but looks so fancy, I love it. Also Octavia Foundation.
If you go to Hampstead, do your research first. We went to the wrong part, walked for miles and hardly found any.
I have a hunch about Clapham - untested so don't shoot me if I'm wrong but I'm planning to check it out when / if we go.