The Queen is a figure head. She is not involved in politics and she certainly doesn't sign new laws. She signed final assenet but she stays far from political debate. If she was opposed she would still sign as she supports the democratic voice of her people. She showed her support in the past and why shouldn't she?? One cant even argue the religious card,the COE supports gay marriage among lay people.
She wouldn't be the first to show support. Victoria attends the LGBT awards. Queen Max certainly does. Modern royals represent modern society and hopefully we will see more open support.
As for if one of her family came out gay, I honestly think she would be supportive, rightfully so.