With the rise of Labour in the 1900's. A possible centrist party of moderate Conservatives and Liberals could be formed and state to be the voice of the nation.
The Unionist party could have been that coalition, but Joseph Chamberlain screwed it up. If you avoid the Khaki Election (which produced a huge majority for the Unionists in that election, but burned a lot of long-term good will with voters) and the Imperial Preference debate (which created a wedge issue that split the Unionist base and made the leadership look weak and ineffective), they may be able to hold a majority for some time.
One potential slip-up would have been Irish independence. Would a Unionist party have allowed it?
Some variation on Home Rule might be possible, if the issue became pressing. Chamberlain drew a distinction between the Home Rule proposal (in which Ireland would have its own parliament for internal matters, but would still send MPs to the London parliament which would govern internal matters in England, Scotland, and Wales as well as overall national matters) and American-style federalism (in which Ireland, England, Scotland, and Wales would each have its own sub-national government). He was vehemently opposed to the former, but at one point expressed a willingness to consider the latter. He's probably a pretty good benchmark for the Liberal Unionists, but I'm not sure if the Conservative Unionists would have supported it.
The problem with this is that federalism was universally unpopular in Ireland - most Irish unionists were diehard intergrationists and even the most moderate of Irish nationalists would have found federalism to fall far short of their desires. This might be irrelevant in getting a bill through (after all in OTL the Government of Ireland Act 1920 was passed with three quarters of Irish MPs abstaining from Westminster altogether) but an Irish federal parliament(s) would have been in crisis from day one.
Why was federalism unpopular with the unionists? It could be good for them.
Why was federalism unpopular with the unionists? It could be good for them.
Slightly different POD (*)
Post WWI.
Anti Coalition Liberals make overtures to the new Labour party to create a formal left-centre alliance.
The alliance wins the 1924 election.
But that's going to create a dominant party system. You'll have a left wing Lib-Lab party and a right-wing Tory party alternating power for the rest of the century. To get a dominant Party system like Canada or Japan you need a single party dominating the centre ground with small, extreme left and right wings groups doomed never to win.