The UK did do this IOTL when Iraq had its 1958 Revolution. The USSR & UK worked to keep Iraq out of the United Arab Republic.
But more widely? No.
Working together is not allying with.
And even having a communist regime will not necessarily turn them into strategic allies with USSR.
China was communist and broke off with USSR in 1959, the 2 countries becoming strategic enemies.
Yougoslavia was communist and non-aligned on USSR.
Whatever their regime, the only reason Britain and France could ally with USSR would be that they and the US have become strategic enemies.
Which is almost impossible because the US holds them financially and has quite many troops stationed in Europe, including in Britain and in France.