How much should I worry about British spelling when submitting to a U.K. journal?
The answer depends very much on the journal. Some expect UK spellings; others will permit either American or British spellings. You should check with the journal in question.

Of course, the other option that you have is the following. Since you know what the major differences between the two sets of spellings are, and you have a sense of which one's won't be caught by your spell checker, you could always do a final search-and-replace after you've completed work on the paper to make sure you've switched everything over. (Or at least, everything you know should be switched over.)

That should satisfy most journals, and as shan23 said, I don't think a journal will reject your paper for writing "meters" instead of "metres"; the most you'll get is a referee report telling you to switch spellings.