Matthew - the legal ownership has passed to your Mother as the surviving joint owner to deal with as appropriate. Probate is not required for the property.
If the property is to be transferred then that would be by your Mother as the legal owner and transferor. As your Father has sadly died he can't be a party to the transfer. You can only ever transfer the whole of the legal ownership
Your Mother plus the two of you would be transferees and identity verification would be need for all three of you.
The application should include a certified copy of either the death certificate or probate
This is covered in several threads on the forum and I always recommend that you do seek legal advice as to what options exist re your late Father's share, your beneficial interest and what's 'best' re the property for both Mum and the two of you.
Land registration is complex, designed to protect legal and financial interests in property. There can be significant consequences for any error. Please consider seeking legal representation. Before making an application without legal representation, it is important to
consider the benefits of using a conveyancer.