London is a large area of course. Much like DC one can go into the areas people by folks making barely a living wage and find great bargains. Or, keep to the expensive areas where bargains can also be found in local markets and such. Further, one can only hang around St. James and such, where everything is "up-scale" in price and quality. But, two blocks away, in a church yard, can't remember the name and too lazy to research it, is a great little market. It was home base for the maker of Milville pipes. He's retired now so that's no longer an option. But, that gives you an idea of what may be just "around the corner" from JJ Fox on St. James, a rather expensive street to shop.
JJ Fox is predominately a cigar store, some Churchill memorabilia on display, pipes and blends available. The staff are fonts of information and it's a great place to get a feel for what is happening in the smoking world of London.
London is what you make it, adventurous or mundane, expensive or not. I like to "base" in the West End (google it) as many great places are within easy walking distance and other places of interest are easy taxi rides.
Hopefully a couple of our London based members will chime in.
Before you leave, research how to recover any and all VAT. It's simple really but, forewarned is forearmed.