This is no ‘British culture war’, this is a few rightwing loudmouths whingeing
People’s attitude to Brexit naturally influences their view of the process of leaving the EU. But outside that, most of the weekly storms and skirmishes that are chalked up to the culture war should instead be understood as rants by bigots and bores. Casting them as some kind of profound conflict makes British society seem more intractably divided than it is, while giving the tedious whingers a political elevation they do not merit.
You keep doing your politics on that basis and the electorate will keep handing you your ass in a handbasket. Culture is in fact the defining cleave in British politics at present.