It's a personal guarantee, so it follows them everywhere. It's got nothing to do with an actual property. And according to Jim the mother will inherit it as joint tenant. And if this is the Dad's only asset, the only way the bank can act on the guarantee is if they put a judgement mortgage on the property.
I'm not sure though if the bank are statute barred after two years in talking a claim against the estate only if probate is not taken out. Or why that makes a difference.
And if the mortgage is being paid off there is no legal reason for the bank to act on the guarantee now. Or even on death. Unless there is some clause in the guarantee of course.