I'd keep a close eye on those bottles. Everytime I have tried to bottle with the Fullers strain it has restarted fermenting and produced gushers with a very bad taste. Ordinarily you'd think that was infection but it's an issue others find repeatedly with this yeast. I keg now and have had no issues with it, but I also learned to always mash low too.
Mine went from 1.060 to 1.014 so I would assume fermentation was mostly complete. No diacetyl either and I used a modest amount of priming sugar (55g dextrose for about 4.6 gallons). I haven't had any gushers and the light carbonation is perfect for the style. It has taken on a slight tartness though, almost cidery as others have described. I'm still enjoying it as the off flavors are far from overwhelming but I would like to avoid this in the future.
My experience is that the flavor becomes more bland as the yeast unfortunately continue to 'clean up' many flavors that are desirable in an english style beer. What you are describing sounds like something else. What was the recipe and what is the 'strange character?'
Recipe is here: http://brewtoad.com/recipes/english-bitter-51. As I said above, after 3 weeks in bottles, it has taken on a slight tartness and sort of cidery flavor. It still retains an minerally/fruity english character though.