I don’t think it’s too tricky?
You just go and apply for a new mortgage just you two, probs with the same lender. Then the solicitor changes the name over with the land registry and then on that you agree to any sum you pay to own half of the property. Then your new mortgage starts (or continue with her existing one but adding your name to it).
I know of some people going through a horrendous divorce because he moved into a property she owned which they then sold to buy their new property together. Shit got MESSY.