First things first, get used to the exchange rate (Currently almost $1.60 per £1)
Most lager (what we would consider as BMC) will run you £2.75-£4 in pubs. That's $4.40-$6.40 per pint of lager (carlsberg, fosters, etc). Drinking quality beer in England isn't always the cheapest affair. But I'd much rather pay £4 for a good craft beer in one of these places than in a standard english pub drinking carlsberg.
1.) Euston Tap (http://www.eustontap.com
A new comer to the craft beer scene in London. Just a couple minutes walk from Euston station, it became my go to place. Great group of guys who run the place, nice tap and bottle selections. Prices aren't too bad but the more rare bottle's can run above £20.
2.) The Rake (http://www.utobeer.co.uk/home.html
Has been and will always be one my favorite places in London. Not a huge tap selection but what they do have will impress. Nice bottle selection with prices being mid-range for both. It's just a few minutes walk from the London Bridge tube stop but worth it. If you can plan you trip(s) here around a trip to the Borough Market
market. An experience that I would recommend to anyone and everyone that visits London. The Borough Market is only open Thursday-Saturday so plan accordingly. Check out Utobeer (owners of The Rake) in the market. They have a nice bottle selection for take away with quite a few rare beers as well.
3.) Cask Pub & Kitchen (http://www.caskpubandkitchen.com/
An amazing selection of draft, cask and bottle beers. If you can't find something to try here that you haven't had before I'd be very, very surprised. I've never left here unhappy and the nice staff and selection make sure of that. The food is quite a bit better than the standard bar/pub food I was used to. Beer prices are mid to high but in the end IMHO worth it.
4.) The Craft Beer Co (http://thecraftbeerco.com/
Awesome! Just flat out awesome! Tons of hand pumps, taps and bottles. Service has always been perfect and the prices are normally mid range. One of the top notch places in London for selection! And you must try their scotch eggs.
There are a couple other places to check out but IMHO they aren't worth going to with the above places available.
And if you have a chance, check out any beer you can from The Kernel brewery. Evin started out as a home brewer and puts out some bad ass beers (IMHO).
If you have any extra time (2-5 days) take the 2hr train ride to Belgium and drink some great beer. I've made that trip 11 times in the past 4 years so if you want some insight to Belgium just let me know.