If property is involved you are going to need grant of probate in respect of the estate of each person involved. That would mean in respect of your mother leaving the property to your father and then your father leaving it to you.
There are 2 “types of probate”.
There is Grant of Probate which the executors need to deal with the estate.
It’s basically a court approval/order allowing them and giving them the authority to deal with the estate assets.
However if there is no will it’s a different document called Letters of Administration although it looks the same (but with a different title) and the effect is the same.
Although the executors apply for Grant of Probate, there are no executors if there is no will and it would normally be the closest living relative. You will find the list of people on the government link I have sent you below.
The government have produced these webpages on exactly this subject:
Please read it in detail.
The property is then transferred from each beneficiary using an assent form AS1 from the land registry.