I could expand on it all a bit, but that could take a post longer than most articles.
Key bits, though:
- No 1922 election; it runs to 1923 as originally planned (as there was no need to collapse the Coalition when they decided not to do Coupon 2: Lloyd-George Boogaloo, because they'd not needed a first Coupon under the new system).
- 1923 sees a Lib/Lab government (after lengthy negotiations), with Asquith PM for one year before stepping down for Lloyd-George (Churchill disagrees with allowing "the Socialists" into government and leads a splinter group of Constitutionalists. It's believed that this is partly because the new system allows for new parties to have a bit more of a chance; this belief is incorrect, because he led a Constitutionalist splinter group from the Liberals in OTL as well). During this time we see a Land Value Tax pushed through against resistance from the HoL (Parliament Act is used). They also extend STV to the county constituencies.
- 1928 sees a reversal in the order of parties and a Lab/Lib government led by MacDonald.
- This means that when the Depression hits (similar to OTL), the Government is one year closer to its scheduled expiry, and MacDonald has a larger "insulation" on his majority (well, he actually has a majority with the Lab/Lib Coalition), allowing him to try some more austere policies before concluding he can't manage it with his mandate and calling an election about one year later than OTL in 1932 (and without opting for a National Government - Lloyd-George has some ideas he thinks the Labour/Liberal group can put into practice. The Government is blamed for the Depression and the Tories win a landslide on their own in 1932. The New Party enters Parliament.
- We can skip through the 1937 election and wartime (WW2 happens on schedule, I'm afraid, and the leader of the Constitutionalist Party ends up PM and there is, as OTL, a Wartime Unity Government.
- We'll also skip through the Commonwealth Party's formation in WW2 and the National Party (Liberals who wanted the Unity Government to continue under Churchill at the end of WW2), the New Party's collapse in WW2, the formation of the New Constitutionalists (partly with the New Party's voters and to Churchill's dislike), the Labour/Co-Op/Commonwealth Government of 1945, with the Liberals entering with them in 1951 but swapping to supporting a Conservative/National/Liberal Coalition in 1952-1954. The Socialist Party enter Parliament to the left of Labour and steal the Communist Party's lunch, as Commonwealth merge with Co-Op and Nationals rename themselves as Reform.
- By the end of the Sixties, commentators asked if we'd remade a 2-party Parliament with the two factions: Conservative/Reform/Liberal versus Labour/Co-operative/Socialist, with the New Constitutionalists sidelined.
- In the turmoil of the Seventies, the Constitutionalists gain more power and the Liberals get sidelined (the right faction shifts rightwards) until Reform (renaming themselves again, this time to "Moderates") collapse the Government as being too right-wing for them.
- The 1976 election sees the most splintered Parliament to date with the two factions weakly cohering if at all; eventually a Labour/Co-operative/Socialist minority Government with Liberal confidence and supply on a year-by-year basis limps into existence and survives until 1980.
- The 1980 election ends up with a Con/Con/Mod win and Labour in recriminations; pulled both left (by the Socialists) and right (by wanting to keep the Liberals on-side as options), they split and the SDP is formed (very similar to OTL; more similar than I liked, but the international economic background and the political unfolding to that point made it a very strong possibility anyway). The defections to the SDP make a sizeable grouping with the Liberals and then the Moderates break away from the right-wing Government (too right wing for their tastes - again. The Tories need to keep the "wets" on side more in this TL or they walk). We now seem to have three groupings: Left (Lab/Co-Op/Soc), Right (Con/Const), and centre (Lib/Mod/SDP). A 1982 Vote of No Confidence brings down the Government.
- The Centre Grouping win a very narrow majority in the 1982 election. One unheralded thing they do is to send a nuclear sub to the South Atlantic when the Argentinian Government start sabre-rattling and quietly let them know about it - there's no Falklands War in this TL. Which has the effect of robbing the Government of the political boost from winning it (whilst saving quite a few lives)
- By 1987, the Tories tack a little centrewards, the Constitutionalists rename themselves "Freedom Party", and the ensuing election allows Con/Free to make a minority Government after tempting the Moderates into confidence and supply.
- The early Nineties are chaotic, with elections in 1992, 1993, and 1996 (the latter seeing the Conservatives suffering by having been pulled left and right trying to keep their minority Government receiving confidence and supply from Moderates, Liberals, and occasionally Freedom simultaneously - and offending every wing going). A wave of defections to both Freedom and the Moderates (who rename themselves the Country Party) mean they're very weak going into the election that they had to call early. A Liberal-led centre grouping is expected to win, but a poor campaign from the Liberals and a very strong campaign from Labour mean that we see a Labour/Co-operative/SDP government (Socialists not invited and Liberals invited but refused - possibly in pique). More Tories defect to the Country Party, who rename themselves to the Rural Party.
- 2001 sees the Labour-led Government re-elected. 2006 sees a slip in share and they have to tempt the Liberals into Government, and then, as the GFC hits in the US, the Government shifts left and tempts the Socialists into Government (arguing that they need a bigger majority). The Liberals get offended and bring down the Government, installing a Con/Freedom/Rural/Liberal administration. Inevitably, the effects of the GFC in the UK are then blamed on Liberal shenanigans (and, somehow, the SDP get a share of the blame. I blame the newspapers).
- In 2011, the Liberals (and, to a lesser extent, the SDP) are smashed in the election. The Rural party rename themselves back to the Moderates (and, by now, there's a running joke of "what are they called today?") The Conservative/Freedom/Moderate Coalition is strong on paper, but unstable over Europe (there's a surprise...). The Mods pull out in 2014 and an election is called - which results in an almost unchanged House of Commons. The Queen quietly lets them all know that they'd better make something work, given that they've asked the country, who've said to keep going with similar numbers of MPs. A Con/Freedom minority limp on to 2016 on a vote-by-vote basis, when by-election losses and defections make it untenable.
- The 2016 election results in a Labour/Co-operative/Socialist minority with confidence and support from the remnant Liberals and SDP. This is withdrawn in 2019, which gives us the third election in five years (what's the odds on that?)
- The 2019 election is chaotic, but a Conservative/Freedom minority tries to govern. This is unstable and collapses immediately after Parliament reconvenes after New Year 2020.
- 2020 election... this is where we come in.
I've glossed over the formations and collapses of various hard right splinter parties that were too right wing even for the New Constitutionalists/Freedom bunch (National Front, of various flavours, BNP, etc); the F3 party is the latest incarnation. And the entry into Parliament of the Communists (just before WW2) and their expiry in the Fifties, or other hard left splinter parties. And the entry into Parliament of the SNP, Plaid Cymru, Greens, and some independents.
I've also not mentioned the effects of the Oil Shock (which happened as OTL in the Seventies, other than the destabilising of politics here), the withdrawal from Empire (always going to happen, but caused the Constitutionalists/Freedom mob to froth at the mouth), Europe (other than the issues in the Nineties), and so on.
In TTL, some commentators insist there's a generational chaotic cycle (the late 2010s, the mid 1990s, the mid 1970s mainly giving rise to this, and then, inevitably, being extended back through the 1950s (when the multi-party factions started to coalesce), the 1930s (weakest chain in the link) and the 1910s (the events leading to STV in the first place and the Con/Lib Coalition, etc). Personally, I think it's just them looking for patterns when none exist.