Your wife's debt passes to her estate and (in the absence of a will to the contrary) that estate passes to you. However an estate can't be 'negative', so you can't acquire a debt directly from your wife's estate.
The building society might (for example) well be interested in things like whether your home was owned by you both as 'joint tenants' or 'tenants in common'. (Ignore if you rent!).
If you were joint tenants then you automatically take over full ownership of the property. However if you were tenants in common then your wife's share of the house formed part of her estate (which she could, if she'd so chosen, have left to someone else). So if (say) your home is valued at £150,000, part of your wife's estate is valued at £75,000 and the lender has the right to demand settlement of the debt out of that money.
It is such matters that the building society will wish to enquire into.