Keith - sometimes only an update re the death as it all depends on what the interested parties expect/want to happen next. But even updating the register re the death is not required until such time as the property is updated for some other reason for example sold
Why is that? Well, we register the legal ownership and the tenants in common/trust/wills all relate on the beneficial ownership. And in many cases when joint owners make such arrangements the register is only part of the consideration following the death as that's already been planned for.
Most tenants in common apply to register a form A restriction to reflect the fact that such an arrangement exists. Updating the register re the death is sometimes all that is required as the restriction protects the trustees/beneficiaries interests as it restricts the sole surviving joint owner.
In some cases you may decide to update the register not only re the death but also transfer the legal ownership from the surviving sole owner to him/her and other(s). That can and does happen but we are not privy to such discussions and we simply update the register.
So what's required is really a question for the surviving owner and the trustees/beneficiaries to discuss and agree. If that was never discussed fully when the tenants in common was created then I would recommend seeking legal/financial advice as appropriate and especially to update everyone's understanding now one of the joint owners has sadly died.
Once a decision has been made and an update is wanted then our online guidance explains some of the most common next steps from a registration perspective