If war broke out, might it be unlikely that fighting would happen in britain?

I think it would be likely.  Britain is a country that is well-developed and has a decent military.  If you are referring to war in London or other big cities, I think the armies would try to keep warfare on the battlefield, not urban areas.  However, to counter that, (correct me if I'm wrong), I believe London is on the water, so a type of naval invasion would be likely.  It also depends on who the fighters are.  Britain v. U.S.A. for example, evacuations of cities would begin, and it's likely we'd see another Refugee Crisis like the one happening now in Europe, but smaller.  But the U.K is part of the U.N., a huge group of unified countries.  If war were to break out, it's likely forces would be pushed out easily with the help of other powers/countries.