I think Twitter's a red herring. IIRC, in the UK the prime minister is chosen by the MPs from the majority party. It's not just the ones on Twitter, though Twitter might be one of the channels some of those members use for rhetoric to influence the final decision. That rhetoric will happen somewhere, Twitter or no.

Also when it gets down to it, in an election the ultimate outcome is decided by the people who are on the fence, which is much smaller that the whole group.