A couple of points;
The Land Registry can't knowingly register a deceased person, so the correct course of action now is to inform the solicitor that your uncle is deceased so he can withdraw the application.
The correct person to make a new application would be the person(s) entitled under his estate. Probate will have to be extracted, and the personal representative will have to assent to vest the property in the new people who would then make the application for ownership.
Unless your uncle is married and/or has children, or his parents are still alive, your father (and any other surviving brothers and sisters) would be the closest relatives for succession purposes since he died intestate.
Once a persons title is barred under the Statute of Limitations, it is barred forever - i.e. once your uncle barred the aunts interest, any interest she had is gone and she would have to retake possesion for at least 12 years to establish any claim to the property. To get to the point of notices being served and waiting for objections, your uncle would have to have shown a decent prima facie case that the aunt is barred, so the fact that he is deceased doesn't change anything for her- if she was barred before he died, she's still barred. Since she didn't live in the property for the last 30-40 years, it's very unlikely that she has any claim.
The big caveat is that with adverse possession, the facts are everything, so have a good chat with the solicitor to go through the case in detail.