Sadly, the ranks of Jazz LP and CD shops in London and much of the U.K. have sadly been depleted over the last decade or so, Bol. Even the flagship HMV store closed up shop, thanks to the Internet.
First port of call would be Ray's Jazz located in Foyles bookshop on Charing Cross Road. Not a huge place but always worth a look, plus they have a decent selection of jazz literature and a really great coffee shop.
Honest Jons in Portobello is always worth a look - particularly if you like Sun Ra, Art Yard etc.
In the suburbs there's the likes of Soul Brother located in Streatham, good for soul jazz in particular.
Never been a great fan of the vinyl places in Soho but some of them are still in operation around Berwick St I think.
Your timing is good for the London Jazz Festival in November - The Cookers and Wayne Shorter are both appearing, there's lots going on by the look of it, worth checking out. Interesting things like Evan Parker at Kings Place, Mike Westbrook solo plus quite a few free events e,g, at Barbican.
Best of wishes with your move !