The following information is taken from The Electoral System in Britain 1918-1951, by D. E. Butler, Oxford University Press, 1953.
The Speaker's Conference on electoral reform reported in January 1917. It unanimously recommended the introduction of STV for parliamentary elections.
In August 1917 the House of Commons rejected the Single Transferable Vote (STV) by 32 votes in the committee stage of the Representation of the People Bill, but substituted the alternative vote by a majority of one. The House of Lords voted in favour of STV, but the House of Commons insisted on the alternative vote. A compromise was arranged whereby the alternative vote was abandoned and the Boundary Commissioners were asked to prepare a limited scheme of STV to apply to 100 seats.
When the Boundary Commission produced the scheme it was rejected by the House of Commons. However STV was introduced for university constituencies.
On 8 April 1921 the House of Commons rejected a Private Member's Bill to introduce STV by 211 votes to 112.
In March 1923 a Liberal attempt to introduce an Alternative Vote Bill was defeated by 208 votes to 178.
On 2 May 1924, with a Labour government dependent on Liberal support in office, the House of Commons defeated a Private Member's Bill to introduce STV by 240 votes to 146.
In January 1931 the Labour government, which was dependent on Liberal support, introduced a Representation of the People Bill which included a clause providing for the alternative vote (AV). On 24 February 1931 this received a second reading in the House of Commons by 295 votes to 230. In the committee stage the clause providing for the alternative vote was carried by 277 votes to 253. The Speaker had refused to accept instructions which would have allowed the discussion of STV.
In June 1931 the House of Lords gave the Bill a second reading. In the committee stage an amendment to substitute STV for AV in a hundred constituencies was abandoned because it was outside the scope of the Bill. An amendment to limit the application of AV to constituencies in boroughs which had a population of more than 200,000 was passed by 80 votes to 29. The Bill received its third reading in the House of Lords on 21 July and returned to the House of Commons.
In August the Labour government fell and no more was heard about the Bill in public.
If legislation had been enacted which provided for about two-thirds of MPs to be elected by STV, the consequence would have been an increase in the number of Liberals elected with a corresponding reduction in the number of Conservatives and Labour. But it would have applied to borough and compact county constituencies, not to large-sized county constituencies where the Liberal Party did relatively well against the Conservatives and Labour was weak. If in those constituencies MPs had been elected by the AV, the Liberals would have been the second choice vote of Labour voters.
It would have been rare for one party to have an overall majority in the House of Commons.