Hi Andrew - I'm sorry to read of your loss.
I'm afraid it's not our role to advise on what has to or needs to happen here as we register the outcome of such discussions/deliberations. And as such
our online guidance then explains how to apply to update the register depending on what you decide.
So it's really legal/financial advice you need here as to what's the 'best' option which maybe do nothing, update the register re the death alone, or transfer the property into joint names again.
In my experience much will depend on how the arrangement (trust) was set up re wills, trust deed etc and whether that then explained what happens next if one of them died. That may help understand what can, needs or has to be done?
Much will also depend on what Father and wife both want to now do and of course how their own health might impact in the future and how best to deal with their beneficial shares etc
So plenty to consider and once done and decided the linked guidance clicks in re what we then need
I'm sorry I can't be more helpful