There are a number of casinos in London, ranging from elegant and expensive to downright scruffy.
You have to be a member, which means you fill out the application, pay the fee if any and wait 24 hours. You must do this at every club you wish to play.
You can go as a guest of a member, but that member must accompany you and must be on premisis with you.
The best clubs are Aspinall's Curzon and Crockford's, both in Curzon Street, and The Clermont Club at 44 Berkeley Square. The membership at each is L500. It's not too difficult to become a meber of Crockfords if you have the 500 quid, but it is a bit harder for the other two. Other high dollar clubs are the Palm Beach just off Berkeley Square and Les Ambassadeurs in Hamilton Place, just off Park Lane.
The Clermont is probably the loveliest casino in the world and is in a listed building, the last surviving William Kent townhouse in London. If you drop by about 2 p.m., when gambling starts, and dress nicely in a jacket (no jeans, tennis shoes, etc.) the club management will likely show you over the building. They won't let you buy any chips to cart away, though.
The most popular casino with Americans is the Victoria on Edgeware Road. Membership is free and the casino is open and somewhat modern (or at least circa disco 70s modern.) It also has a poker room, one of only two casinos in London so equipped.

Cheers,
Russ