The Cabinet is gathering now at #10, looks like a resignation demand is in the works. It's interesting that there is a not an obvious successor waiting in the wings; it's going to require much jostling in the proverbial smoke-filled room to determine who will replace him if he goes.
As to the rules/terminology being different than in the US, it's rather the other way 'round: the American system is the one that is unusual. The UK uses the standard Parliamentary system, wherein members of the public vote for their local member of Parliament, and the major government positions are then meted out by jockeying within the majority party, or with the formation of party coalitions.