Where do I start? "British" (Scottish, English, Irish, and Welsh) ales are my thing. As mentioned previously, I love using WLP002/Wyeast1968 (Fullers) for the definitive Fullers taste - if you don't want that flavour profile, don't use it. :-)
The other ones I really like are WLP023 Burton Ale, supposedly the same strain as Wyeast Thames Valley Ale although I'm not personally convinced. I'd have to do a side-by-side comparison. Regardless, I love both of those yeasts for a variety of malty or hop forward British styles. They are not necessarily "clean" as in Chico-clean but they aren't the Fullers flavour either. The Burton does have a bit of fruit, pleasantly. Kristen believes them both to be the Brakspear strain.
As also mentioned, I love Wyeast West Yorkshire (allegedly the Timothy Taylor yeast) but it does have a lot of character (for me, in a good way, but if you're trying to avoid that...). If you've had the pleasure of tasting "Landlord", or, I'll throw in Black Sheep Ale, Wychwood Hobgoblin, or Theakstons XB, as they are similar in brewing technique/character and you like that flavour profile, use the West Yorkshire. If you do, keep in mind that the prototypical method of fermenting with a "Yorkshire" yeast is with a Yorkshire Stone Square system. I won't go into details, but simply, to get the most out of this yeast, ester-wise, consider recirculating/rousing the yeast in your fermenter every few hours for the first few days. (It goes without saying, maintain sanitation).
Two very specific yeasts I can recommend: available all year is Wyeast Northwest Ale and the seasonal White Labs Bedford Ale Yeast. Northwest Ale has a confusing but valid name. The strain allegedly originated at Gales Brewery in Horndean, England but is used by Hales in Washington state. Gales was purchased by Fullers a number of years ago and a few of the original Gales beers are now brewed by Fullers using the Gales yeast. Gales HSB is an incredible and quite unique ESB style beer but you can only get it in the UK, sadly. The Northwest Ale strain does a great job at replicating its flavour - it is quite malt forward. As for the Bedford, I can state first hand, if you want to brew a Well's Bombardier clone, you cannot do it without the Bedford Ale yeast, again seasonal from White Labs. Listen to my appearance on Jamil's "Can You Brew It?" for Well's Bombardier for more details.
Incidentally, I did the interviews for CYBI for Fullers, Black Sheep, Wells, Wychwood, Meantime, and my only US brewery, Ska, as well as discussing the Fullers Parti-Gyle method on a recent Brew Strong episode. So, yes, "British" is pretty much my thing.
Those are the "British" strains I've used and have liked. The others, I have used but have not been enamoured with - excepting the seasonal White Labs "Essex Ale", I want to try it. Oh, and finally, I have friends, and even Pro brewers (in the UK) who have used dry Nottingham and have been very pleased. I've also had great results from Fermentis S-04 but I'd highly recommend hydrating either strain before use. I can tell you, S-04 is one of, if not, the best dry ale strain as far as quick primary fermentation and beer clarity. That sucker really packs down in the bottom of the fermenter and gives you a very clear beer. My only gripe is I've had a few beers turn out too estery with it, and not necessarily in a good way. However, don't take that to heart, this was years ago and I would handle the yeast quite a bit differently now - I guess it's time for a Brulosophy (Marshall) experiment. :-) [Highest respect for that man and his compatriots at the website/blog].
My two pence. Hope that helps, not hinders. :-o