The lender would be not be happy with this as the mum would still living in the house. Most lenders these days would insist on vacant possession and even if they were willing to do this then the op would need to buy the house from her and would have to pay legal costs and stamp duty if sale is...
The only way to take over the house would be to buy it off your gf's mum. But then you will have pay stamp duty, solicitors etc. The lender will not entertain changing names on the mortgage if she is 4 months behind. The best thing you could do would be to lend her the money and place a charge...