Katy - wills deal with the wishes of individuals. The legal ownership, which we register, does not transfer by virtue of a will/their wishes alone. To transfer the legal ownership it has to be transferred as a whole and in your scenario that would be by your Mother to herself, you and your sister if that's what is wanted.
Your late Father's half share, and indeed your Mother's half share, are in the beneficial ownership. That can be split/shared but the legal ownership is always dealt with as a whole.
Before transferring the legal ownership please do seek legal advice. Understanding not only the difference between the legal and beneficial ownerships but also the impact of being a legal owner and/or what may happen with the beneficial shares in the future may impact on what you do next re updating the register and the legal/beneficial ownerships.
Legal and Beneficial Ownership · HM Land Registry
We register the outcome of such considerations so are unable to advise on what's 'best' in such scenarios but once next steps are decided we can steer you to the guidance/forms needed to update the register