In the 11 by-elections from January to August 1939 in seats where Conservative and Labour where in first or second place, there was an average swing from Conservative (and other parties supporting the so-called 'National' government) to Labour of 4.1%. In the 4 by-elections where Conservative and Liberal were in first and second place, the average swing from Conservative to Liberal was 4.8%. The Westminster, Abbey by-election on 17 May was a straight fight between Conservative and Independent Progressive. It was held by the Conservatives with a drop of 10.1% in their vote compared with the 1935 general election which was a straight fight between Conservative and Labour.
In the 1935 general election Conservatives polled 53.7% of the national vote to 37.9% for Labour. So Labour would need in the region of a 6 to 7 percent swing to win an majority of seats in the House of Commons. In the 1945 general election there was a national swing of 11.6% from Conservative to Labour.
However the Oxford by-election on 27 October 1938 and the Bridgwater by-election of 17 November 1938 were straight fights between Conservative and Independent Progressive candidates supported by Labour, Liberal and some anti-appeasement Tories. In Oxford the Tories held the seat but with a drop of 6.7% compared with 1935. Bridgwater was gained by the Independent Progressive candidate with a fall of 10.1% in the Tory vote.
So in a general election in the spring or summer of 1939 an anti-Chamberlain alliance of Labour, Liberal, dissident Tories( such as Churchill, Duff Cooper, Eden, Macmillan, Brendan Bracken, Robert Boothby) and left-wing independents could win a majority in the House of Commons. With Labour as the largest party in that alliance, Clement Attlee becomes Prime Minister. He forms a coalition government of Labour, Liberal and dissident Tories. He appoints Churchill to his cabinet, and probably Duff Cooper and/or Eden.