Graham - if the legal ownership is to be transferred then the surviving owner would do that as Transferor. They would transfer the whole of the legal title to themselves and the new joint owner(s) as appropriate Please do seek legal advice before making any changes as there are options, including making no changes to the register, and it's worth discussing how the trust was set up, protected and what might be 'best' re next steps.
HMLR Guide: Start - External · HM Land Registry
Note - probate was not required re the property as the legal ownership did not form part of the joint owner's estate. The TIC aspect you refer to relates to the beneficial, not legal, ownerships
Similatr scenarios are raised and responded to on this forum and might help also
