It depends how the house is owned. Assuming the house is in both names, if they own it as 'joint tenants' then both of them own the whole property. When one dies, then the survivor automatically owns the lot.
If it is owned as 'tenants in common' then each own a share - a half each, as an example. When one dies, his half forms part of the estate and can be willed to anybody.
So, if they own the property as 'joint tenants' then the house does not form part of the �125,000 rule you refer to and the children will not have a claim. If she dies first without a will, everything will go straight to the husband.