forget the mortgage for the moment.
Mr and Mrs A own a house together without a mortgage.
as part of their separation agreement, Mr A moves out and Mrs A stays in the house.
It may stick in Mr A's craw that Mrs A's new partner has moved into "his" house, but I can't see scope for action here.
He could argue that Mrs A now has additional rental income from a tenant, and so the separation agreement would need to be revised.
They own the house jointly, so Mr A could move back in I suppose.
Are there children involved? Is Mr A paying maintenance? what are their respective incomes?
As MF says, this is not really a legal problem. It is a practical problem but the whole separation needs to be sorted out. Financially, it may suit Mr A that there is a tenant in the house. But the emotional issues outweigh the financial.
now add in the complications of the mortgage
Is the house in negative equity?
If it is in negative equity and Mr A is paying nothing, then he has no moral or legal right to object, I would imagine.
Finally, what does Mr A's solicitor advise?
Mr A's solicitor would know the full history.
I suspect that they have given the same advice as MF1 and Mr A doesn't want to hear it.