The children are not an issue then and hence, it’s unlikely she could resist an application to court for an order for sale.
Debts only die with a person if there are no assets. Here, there are assets in the form of the house andhence, some of the house proceeds would have to go to pay the debts.
What happens to the pension would depend on the terms of the pension scheme. He may have appointed his partner as beneficiary or left it open. Sometimes the benefit dies with the deceased, sometimes it goes to children or sometimes to a legal spouse. It would be necessary to contact the pension scheme to find out whether your late father opted for anything and if so, what that was. The pension scheme administrators are going to need a copy of the death certificate.
There is no central register of Wills although there have been many attempts to get one together. It is a case of ringing round all the solicitors or writing to them asking them whether they hold a will. The exercise is not very often successful.
You can enter a Caveat at the Probate registry would give you notice if anyone applies for probate on a will.
There is no need for probate if the property is in joint names as joint tenants.
The comment you make about it being unfair for her to inherit the whole house (under the right of survivorship) and then leave it just to her children is just the way the law works.
Your late father could have got round this (too late now) by writing a will and making sure that all the beneficiaries were aware of its content and where it is or at least, making sure that the executors were aware of the Wills existence and its whereabouts.
You might want to start with getting the deeds of the property by following this link
they only cost 3 pounds. You will then know what names the property are in and I can tell you, if you post the deed on here, whether it’s joint tenants or tenants in common and whether she is entitled to the property or whether probate must be applied for.