This is a difficult question not only because there are so many great British yeasts available these days, but there is also a general lack of understanding of how to best use these yeasts. Most British strains work best when fermented via a particular process and it can be difficult to get good results when most home brew fermentations are the same for every yeast. Also, American expectations of British yeast flavor is often different than what is found in the UK. I tend to prefer cleaner and more restrained character; yeasts that are overly tart, fruity, and diacetyl heavy are my least favorite.
That said, here is what I like:
WLP006 Bedford Bitter: Probably my favorite, certainly for bitters and hoppier styles. Great malt character, with mild fruitiness, and balance. Lets all ingredients shine. Same yeast is used via CCV and does well with moderate 02 and low ferment temps. Not top cropping and the yeast can be powdery.
WY1318 London III: Excellent for milds, browns, and maltier styles. Temperature tolerant and not a big diacetyl producer. Mild fruity character that goes sweet. Best used as top cropper and will make very clear beer when done right.
WY1882 Thames Valley II: Great for malt/balanced styles, will throw some D, but cleans up very well. Similar to WY1968 but cleaner and slightly less flocculant. Makes great mild, brown, and ESBs.
WY1028 London Ale: Probably the best yeast out there for London style porters and big stouts. Dry, slightly minerally, little to no D, and temp tolerant.
WLP022 Essex Ale: Dry, slightly bready, and lightly fruity. Can make clean, hop forward bitters, pales, IPA, golden ales. Nice top cropper.
WY1098 British Ale: Can be an easy to use and forgiving yeast, although you need to keep the fermentation temp low or suffer from tart and tangy flavors. I do not care for the flavor of this yeast at all, but a lot of people like it.