Did any element of the Aristocracy/House of Lords or the Monarchy run to the tunnels? Because they might form alternative centers of gravity to the PM and the House of Commons - or at the very least, inspirations to other factions. Some sort of ultra monarchist faction, perhaps. And, unfortunately, probably some sort of BNP/Far-right inspired faction too, though we could just ignore that for the sake of variety, since it would basically be a British version of the Metro Fourth Reich.