As they obviously have been given a mortgage, then they should just go along with it.. Provided they, somehow, manage to meet their monthly payment, the Mortgage company need never know that he is not in regular employment at present.
Eventually, when he gets another job, they just then need to inform the mortgage company - but do not think they will be particularly interested provided they receive the money each month.
I can remember,many years back, of something similar happening to me and my hubbie,- but he lost his job BEFORE the mortgage offer was made. I can really, even now - nearly 50 years later, remember my feelings of total desperation and asking people to help. In the end hubbie did get another job in time for us to make proper application, etc.
Do hope we get a happy ending for this