Depends on how her retirement savings are set up — the rules of the arrangement may operate to pay the money directly to your father as her nominated beneficiary, in which case there is no need — so far as the retirement savings are concerned — for a grant of probate. As MHoran379 says, ask the administrators of the arrangement.
Similarly, State Savings will tell you if they need you to take out a grant of probate before they will pay out. They are entitled to require this, but most financial institutions have a (relatively low) amount that they will pay out without a grant in a straightforward case, which this appears to be. But the only way to know for sure is to ask them.