There were 670 seats in the British House of Commons before 1918. In general elections from 1885 to December 1910 between 81 and 85 Irish Nationalists were returned at each election. That leaves 585 to 589 seats to be divided between Conservative and any allies, Liberal and Labour. In a theoretical, but very unlikely, equal three party split Labour would be the largest party if they win 196 or 197 seats.
Labour would have to do roughly as well as it did in the 1923 general election in OTL. The result for that election was Conservative 258 seats, Labour 191 seats, Liberals 159 seats and others 7 seats to make a total of 615 seats. That election was Labour's best result to date.
in order to reach the minimum necessary to become the largest party in the House of Commons, Labour would have to break out of mining seats and other heavily working class areas, and win in mixed working class/lower middle class seats.
In OTL 3 Independent Labour MPs were elected in the 1892 general election, out of 9 candidates (
http://www.en.wikipedia.org/wiki/United_Kingdom_general_election,_1892. These were all lost in the 1895 general election.
The Liberals split in 1886 over Home Rule for Ireland. The Liberal Unionists were a combination of the right-wing of the party and Chamberlainites concentrated in the Birmingham area. There would need to be a major issue which would lead to a left-wing split from the party sometime between 1885 and 1900, with the Liberal left-wingers coalescing with Labour.
If the Conservatives win a majority in the 1885 general election, there would be no Irish Home Rule Bill in 1886. If Gladstone resigns as Liberal leader in 1885, the two most likely contenders to succeed him would be Spencer Cavendish, the Marquis of Hartington (at this time a member of the House of Commons), and Joseph Chamberlain. Both men were opposed to Home Rule and became Liberal Unionists in the 1886 Liberal split.
A possible scenario is growing demand from the left-wing of the Liberal Party from 1885 onwards for Irish Home Rule. A possible leader for the left-wing Liberals is Henry Labouchere.
The Conservatives are defeated in a general election in 1891, and the Lberals win a majority of seats in the House of Commons. In that election the official Liberal Party policy is not to introduce legislation to give Home Rule for Ireland, though about 20 to 25 percent of the party supports Home Rule.
By now Hartington has become the Duke of Devonshire and is a member of the House of Lords, so presumably is no longer Liberal Party leader. Chamberlain is the leader and becomes Prime Minister.
Chamberlain does not appoint Labouchere to his government. Labouchere and the left-wing/radical Liberals introduce a bill in the House of Commons to give Home Rule to Ireland. It has the support of 70 to 80 Liberal MPs and the Irish Nationalists but is defeated by the votes of most Liberals and the Conservatives.
These Liberals become increasingly dissatisfied with the Chamberlain government and its imperialist policies. By the next general election in 1896 or 1897 they have formed themselves into a new party, let us say called the Radical Party. They fight the election in alliance with the ILP and the two parties together win say between 20 and 60 seats.
There is a steady drift of Liberals to the Radical Party. This is accelerated in the early 1900s when Chamberlain splits the Liberal Party over tariff reform and imperial preference. A minority follows him and joins with Conservative tariff reformers. A significant number of free trade Liberals join the Radicals, while in the middle a shrinking number of free trade Liberal remain in the party, though bolstered by a small number of free traders crossing over from the Conservative Party.
Sometime in the 1900s the Radicals and Labour merge to form one party, perhaps called the Radical Labour Party (RLP). With the Liberal Party in electoral decline, the RLP wins either a majority of seats, or is the largest party, in a general election sometime between 1908 or 1909 and July 1914 and forms a government.