There'se nothing unBritish about the spelling 'apologize'. It has been the house style of The Times for well over a hundred years, and is used by many large and influential publishers (Oxford University Press, for example). I'm tired of being accused of flirting with modernity and excessive American influence, just because I use a spelling that millions of British people use (so long as they haven't been got at by generations of school-teachers peddling misinformation).
In some cases (very few) '-ise' needs to be used instead of '-ize' in Br Eng spelling; and (invariably, I think) '-yse' instead of '-yze'. AmE ironed out these exceptions; so Z spellings are characteristic of AmE. But they're not exclusively American. (I can explain the exceptions if you want, but I think people's eyes would start to glaze over.)
Meanwhile, back at the question...