You do not say where you live and frankly how you deal with this situation does entirely depend on what the law is where you are.
Few on Gransnet ever stop to think that both property and inheritance laws are different in English and Scots law, and if, as many gransnetters do, you live in Canada or Australia or anywhere else, most of what has been stated in answer to your question is probably of no use to you.
Your friend needs to consult a solicitor first thing. He also needs to try to talk to his mother about doing as her father's will stipulates and handing over his and his sister's part of his estate.
It will not be an easy or pleasant talk, but if he starts by telling her that he and his girlfriend are considering marriage, children and their own home, that might just ease the situation. Most women her age are looking forward to grandchildren. He also needs to try and convince his sister that the present situation cannot continue.
If it does, I mean if he and his sister take the line of least resistance and let their mother stay on in the house, how do they know they ever will get their inheritance? If she cannot get a mortage and cannot afford to buy them out, can she afford the necessary repairs to the house as time goes on?
What do they do, if she suddenly wants to sell her half of the property? What indeed do they do if she suddenly drops dead?
This man and his sister need sound legal advice NOW.
They have wasted time and probably money as well on "doing up the house" - stupid thing to have done, as it will not increase the price the y get if the house is sold. Don't waste more time.