Don't change your spelling, but do get a Brit-picker. If you do decide to change your spelling as well, make sure that your Brit-picker is thorough to the point of OCD, because while American spelling and British terminology is fine, British spelling and American terminology is not.

Also, remember that the UK is actually made up of four very different countries. England, Scotland, Wales and N. Ireland all have their own identities and idiosyncrasies. If your story is set in Wales, get a Welsh-picker. If you're writing from a Scottish character's POV, get a Scot-picker. English people tend to be woefully ignorant about the other cultures around them.