Apparently it is fully supported now
https://www.lsu.edu/faculty/lray2/teaching/4068/cpb.htm
“Gay Marriage
The restriction of homosexual rights is yet another issue that the Communist Party wishes to rectify. As of December 5th homosexuals wishing to make their relationships official now can register these relationships. These will be recognized not as marriages, but as �civil partnerships�. In civil partnerships, gay partners can claim each other as next-of-kin, receive each others� pension, and avoid inheritance taxes on goods passed. This is a progressive step; however it is not enough in the movement towards equality in our society. Committed homosexual couples should be allowed to have every opportunity that heterosexual couples enjoy, which include adoption, in vitro fertilization, and monetary rights.
Homosexuals should be given the right to marry, as heterosexual couples are allowed to marry. Marriages should be a legal relationship regulated by the government; the Anglican Church should have no say in such governmental policies. As religion, which has no place in our government, has become less of an important issue to the people of the United Kingdom, their acceptance of the issue has increased. The populace of the United Kingdom appears to be moving away from the traditional Anglican view of denying homosexuals their rights.
The Communist Party will give equality to all people of the United Kingdom; we will establish a system that does not exclude any minority groups. In regard to this issue, the Communist Party will legalize homosexual marriage in order to foster the national equality at the heart of our party. Equality is necessary for the development of a state, and homosexuals should not be excluded from this development of our society.”