I'm not certain what the UK's process is in the event of a resignation.
The Conservative party runs the country. If the PM resigns, the Conservatives just pick a new leader from their own ranks, who then becomes the next PM. They don't have to go to the electorate.
What should we expect from the new Prime Minister? or is he likely to resign as his predessesor?
It's complicated. He's much more competent, so less likely to make calamitous blunders where he'd have to resign. The big question is whether his party just implodes.
I would be interested to know what the british think about all of these recent developments. Even if I had no opinion on #brexit it was still interesting to see what brits had to say about it. Them not being as polarized or partisan as americans trend towards being.
I'm British, so can offer an opinion from that perspective (but of course it's just my opinion, and others would no doubt disagree). Apologies for the wall of text, but hopefully you might find some of it illuminating.
If we want to understand the current situation, then we need some background. 2010 is probably the best place to start...
Politicians are to an extent judged on events that happen around them, regardless of how well or badly they perform. Labour (left of centre) were in office at the time of the global financial crisis, and when the next election came around in 2010, the Conservatives (right of centre) ran an election campaign blaming Labour for the crash. The campaign was effective, and the Conservatives won and took control of the country. They then began a process of austerity, drastically cutting public funding across the board. I'd contend that this approach was largely ideologically driven rather than through necessity. The crash gave them an excuse to cut services and reduce the size of the state, to do what they'd always wanted to do but had previously shied away from due to its unpopularity. Now they could claim that they were simply being economically responsible, financially prudent, and that there was no other option.
The inevitable consequence of Conservative rule is that money gets taken from ordinary people and given to the rich, and inequality increases. The effect was exacerbated this time around due to the unprecedented levels of austerity, and many people, even those in full-time employment, began to struggle to survive. Food banks proliferated. The Conservatives and their friends who control the media then just relied on their tried and tested divide and conquer strategy. If you're poor and struggling, it's not your fault, and it's not the government's fault, it's [insert sub-group of the powerless] who are to blame. They decided to blame European immigrants.... the Poles and Romanians are taking your jobs, they're exploiting EU freedom of movement to use our health service for free and then go back home without paying anything, they're the reason you're poor, the reason that services are suddenly so over-stretched, that it takes you weeks to get a doctor's appointment, that you can't afford to buy a house, that you are suffering. We want to help you, we want to stop them coming here, but we can't because Europe won't let us. It's Europe's fault.
As a consequence of this media offensive, support for the small, obscure, anti-immigration, anti-EU, UKIP (UK Independence) party began to rise dramatically. UKIP threatened to suck up right-wing votes, which would hurt the Conservatives in an election, and potentially let Labour back into power. So the Conservatives needed to appeal to the xenophobic tendencies that they'd helped to promote. And the threat wasn't entirely external, there had always been a section of the Conservative party that were hard-right zealots intent on leaving Europe in order to be able to remove EU-imposed protections on workers' rights, sick and redundancy pay, environmental standards, etc. So the Conservatives, as well as heading off the threat of UKIP, also needed to heal the developing rift within their own ranks.
The eventual result was that the Conservatives
promised the public an in/out referendum on the EU question, and due to giving this promise they successfully held off the threat of UKIP, won the next election, and retained power. Next came the EU vote. On one side of the debate, the perfectly clear economic argument of remaining a part of your largest market. On the other side, xenophobic hysteria. The prime minister (David Cameron), knowing that he himself, most of his own party, and almost the entirety of the opposition Labour party, wanted to remain a part of the EU, assumed that the referendum result would be fairly clear-cut in favour of remain. This was a catastrophic miscalculation, as leave won the vote (52% to 48%), and the UK was suddenly committed to leaving the EU. The prime minister duly resigned, abdicating all responsibility for what he'd done, and leaving someone else to sort out the mess.
This is a large part of why we have had so many prime ministers in such a short space of time. The Conservatives have to be seen to support the referendum result. A largely pro-remain party has to become pro-leave, and deal with the aftermath. The supposedly pro-business party has to throw up barriers to trade. They also have to solve unsolveable questions, such as: where should the border between the EU and the UK be? The Republic of Ireland remains part of the EU, but we can't have a hard border between Northern and Southern Ireland for obvious historical reasons, and we can't have a border in the Irish Sea, because that would split Northern Ireland off from the rest of the UK, and would be seen as surrendering the territory to Europe. Basic, simple questions, that the people who voted leave either couldn't comprehend or just disregarded because it wasn't as important as getting rid of the foreigners.
These issues have for the last few years been threatening to tear the Conservative party apart. Many of the Conservatives are pro-EU membership, but must be seen to be pro-leave. Meanwhile the pro-leave faction are steadily gaining control of the party. It's civil war in all but name. And then came Covid, and a new global economic downturn. The world has to fight to recover, but the UK has to recover from both this and Brexit.
Our populist, foreigner-bashin' Trump-lite prime minister, Boris Johnson, was forced to resign after a series of scandals. The Conservatives command a sizeable majority in parliament, and for so long as they can muster enough votes from amongst their own ranks to keep the government functioning, they don't need to call an election until the end of 2024.
After Johnson resigned, there was a Conservative leadership contest, which would give us our new prime minister. This contest had two parts. Firstly, Conservative MPs selected their favourite candidates from their own ranks, and through voting whittled this down to a final two (Rishi Sunak and Liz Truss). Sunak was the overwhelming favourite amongst MPs, but the leadership contest has a second part... once it's down to the final two, then members of the Conservative party (not MPs, rather that small section of the general public who are members of the wider party) vote to determine who wins from the final two. There was absolutely no chance that those racist bigots would vote for the one with brown skin, regardless of the fact that Sunak was by far the more qualified of the two... so they elected Truss, who became the new PM despite the MPs of her own party not wanting her, and her not being elected by the general population. We ended up with an imbecile in charge, with inevitable consequences, a disastrous unfunded mini-budget, market chaos, recrimination and finally resignation and another leadership contest to elect yet another new PM. Similar rules to last time, first the MPs pick their top choices - not the final two this time, just anyone who gets over 100 Conservative MPs supporting them - and then the party membership vote for their favourite. And the result was that Sunak was the only one to get the 100 nominations, which meant the MPs didn't need to go to the membership for the second round (if they had, it would surely once again have been a resounding win for 'anyone but the brown-skinned guy').
The result is that Sunak is our new PM. He has the backing of most of the Conservative MPs, but there is a sizeable minority who blame him for the downfall of Boris Johnson, and who might vote with the opposition and so limit what he can achieve in parliament - they could conceivably make the government entirely powerless if they so wished. And because Sunak has brown skin and is (gasp!) a Hindu, many Conservative voters are outraged (we voted for Brexit! we voted to get these people out!*). The Labour party and other opposition parties are saying we need a general election, because we now have yet another PM who has no mandate. Sunak doesn't want an election, because Truss destroyed support for the Conservatives, and they will undoubtedly lose badly and relinquish power to a resurgent Labour party. It's up to the Conservatives really, if they can hold themselves together, they have no need to call an election until Dec 2024. The issue is whether they can hold themselves together and vote together as a functioning government. Now that Sunak has taken over, we are for the first time starting to see some Conservative voters wanting an election just to get rid of the brown-skinned guy, some Conservative MPs wanting an election to get rid of the guy who helped bring down Boris Johnson, and some Conservative MPs who, more strategically, can see that their party is in a complete mess, and would rather give up power for a few years, let Labour in to take control and make the painful decisions to get the country running again, and then try to get back in at the next election by blaming Labour for the pain of fixing everything.
*
The fact that xenophobes and racists are largely the same, and that the xenophobes who wanted us to leave the EU did so in large part to get rid of people from India and Pakistan, despite India and Pakistan not being a part of Europe, says a lot about the intellectual wattage of the electorate and the wisdom of holding a plebiscite on anything important.