In initial trade, London's benchmark FTSE 100 index advanced 0.4 percent to 6,830.81 points.
In the eurozone, Frankfurt's DAX 30 won 0.5 percent to 10,832.04 points and the Paris CAC 40 climbed 0.6 percent to 4,834.24.
The pound slipped as low as $1.2478 as the head of a Conservative Party committee in charge of organising leadership challenges announced that May will face a no-confidence vote.
The chief of the so-called 1922 Committee, Graham Brady, said the vote by Conservative members of parliament will be held between 1800 and 2000 GMT, with the result announced shortly afterwards.
"The question for the pound is not who the leader is per se, but what it means for Brexit," said Markets.com analyst Neil Wilson.