Not sure on the rules of surnames!

So I live in the UK and my partner isn't the father of my baby but it's just aswel he was, the real dad has absolutely no interest whatsoever, my partner has been so amazing throughout the whole pregnancy, I was very early into it when I got with him, anyway he's been at all scans and appointments and he's bought everything and talks to my belly and talks about the baby like its his and I'd love the baby to be able to have his surname, but I don't know if it's allowed, any advice would be great?