I don’t know of the behind the scenes machinations in past cases*, but if they’re convicted of a sufficiently serious crime, like any other member they would be subject to an automatic recall petition (and all bets would be off as to whether other parties would field candidates against them in a by-election, most likely all parties including their own would do so).
But more likely, in such a case, and likewise if they lose mental capacity to do the job, I don’t doubt a precedent and procedure would quickly be found to get them out, either by simple pressure on them to resign or a formal vote of confidence or some such.
Unlike in the US, the Speaker of the House is supposed to be a neutral, non-partisan figure (altho’ the last two ones have not had a good reputation in that respect). He/she is normally chosen from each of the two major parties alternately (altho’ the Labour members supported choosing the present one specifically to annoy the Conservative party) so the next one is likely to come from Labour.
That’s the point. Party loyalty is strong, so the party can nominate a squashed apricot, and be confident that it will be elected. Epping Forest has never, since it was first created, elected any but a Tory MP. That means that whoever wants to be MP for Epping Forest just has to appeal to the local Tory selection committee, not the voters. Whoever gets the nod from the selection committee will be MP for Epping Forest, and will remain MP for Epping Forest from election to election unless they manage to piss off the selection committee to the extent that the committee refuses to nominate them. The result is that the MP can treat the actual voters with blithe disregard.