You should write to Equiniti, sending a certified copy of the death certificate, together with the original Grant of Probate (or a sealed office copy), or the Letters of Administration. Alternatively, you can call Equiniti, to report the death of a shareholder and speak with an agent directly to discuss the steps required.
The name(s) of the deceased person(s) personal representative(s) will be put on the share register while the estate is finalised. Equiniti will then place a temporary marker on the register to ensure that wherever possible, they do not issue further dividend payments or other correspondence in the shareholder's name. They will also send you the forms you need to complete.