"Is there an equivalence of flip-flopper in British English or is it used in British English?". Are you referring to just in the UK ? I can speak a little for Australia :) Here you tend to use it more as a verb than a noun. For instance, rather than say someone is a flip-flopper you would more likely say "they flipflop" on a certain position. The conversation gives the context in what they always reverse their opinion on. Otherwise you add a few more words. I do hear it on occasion as a noun, but less and usually this is in a political context "He/she is a flip-flopper !", meaning they (politician) change their political opinion on a certain subject quite often.

May be "inconstant".