Sorry to disappoint you, but London - one of the world's most populous cities - is really poorly catered for with wargames shops. Most Brits buy their figures online or at wargames shows.
Even Orc's Nest is not that great to be honest - it carries most (but not all) Copplestone, a smattering of Westwind, and various fantasy ranges, plus lots of rules, boardgames etc. But that's about it. There used to be several good wargames shops in London - now there are none. Apart from about 100 GW stores, obviously. Perhaps there's a connection
To be fair though, there used to be very few wargames shows - now there's one almost every weekend somewhere in the country - and that's where people buy stuff.
Shame you're coming in May not April, or you could have hit Salute and shopped til you dropped
There is Leisure Games in North London, which I've never been to, but lots of people seem to recommend. They say they carry Artizan and other ranges, but worth checking before you visit to see what they actually have in stock. (
http://www.leisuregames.com/index.html).
My local wargames shop
claims to stock Artizan, Copplestone, Westwind, Perry, and various others - and usually has about 3 blisters from each manufacturer in stock
4D Modelshop is a terrain builders paradise, and a 10 minute walk from the Tower of London - so worth a look if you make terrain and you're taking in the tourist sights.
On the wider front, I'd certainly recommend a visit the both the National Army Museum in Chelsea and the Imperial War Museum in Kennington. Both free to enter.
But shops selling wargames figures? Meh. Nothing much doing I'm afraid.