In many states, this would be about where the army steps in ... except that the UK has (since Cromwell) never suffered much from the "man on a white horse" problem (they've had one general become PM, and Wellington was sort of a special case; compare the US (Washington, Jackson, Harrison, Taylor, Grant, Eisenhower)) and their military culture is dead set against that sort of thing.

I really have no idea of how they get back from here, or how they go forward. The institutions of control - police and taxation and regulatory bodies and so forth - are still functional, but Brexit isn't the only scandal that's been eating away at the perceived legitimacy of the governing élites (Windrush, Grenfell Tower fire, failure of Universal Credit, NHS problems). The most likely immediate effect at the government level, at a guess, will be a splintering of political support to minor parties, creating a series of hung parliaments and general legislative paralysis, with a host of really critical issues to deal with looming. Beyond that... I don't know.