June - any conveyancer will charge a professional fee to cover the services offered etc
From a purely registration perspective you are purchasing part of the freehold so a form TP1 with suitable plan will be required - see
Registering land or property with HM Land Registry: Change the registered owner name - GOV.UK re forms and
Preparing plans for HM Land Registry applications (PG40s2) - GOV.UK section 3 for quality of plan needed for a Transfer of Part.
I don't know what 'action' you might be thinking of but if you are concerned re the professional fees being quoted then you can shop around for quotes if necessary