The mortgage debt will be 'joint and several'. That means that the lender has the right to pursue either or both of the debtors for any outstanding money. So if, for example, the man was to leave his partner (with her remaining in the house) and the mortgage payments fell into arrears, the lender could choose to pursue the non-resident man alone if they thought that he was the one who would have the most money.
Similarly, if one partner ceases to pay their share of the mortgage because they've died, the lender can expect the surviving partner to shoulder the burden of paying the full amount.
If the couple bought the property as 'joint tenants' the whole of the property will automatically belong to the surviving partner (with the deceased partner's son having no claim to any part of it).
If they purchased it as 'tenants in common' the son will acquire ownership of half the house (but with the surviving partner remaining solely responsible for the mortgage repayments).