Hi Hattie - whilst you don;t have to use a conveyancer I would always recommend doing so as they can help with legal as well as registration advice.
From what you have posted I assume that the property was registered in both names. If so then probate for Mum isn't needed as the legal ownership passed to your Father on her death. So if the property is to be transferred it would be by him and forms AP1, TR1 and ID1 are the right forms. We'd also need an official/certified copy of Mum's probate or death certificate to confirm her death.
So the key consideration seems to be the power of attorney you have and what supporting evidence re his mental state would be needed to enable you to act his attorney and transfer the legal ownership to yourself. This is where your legal advice and assistance needs to come in and should, in my experience. also cover whether that's the 'best' thing to do for both him/you
Whilst we can
offer registration guidance re powers of attorney and how they might be used in support of a transfer/application we can't advise you on the law and the options to be considered