As deaths are now registered over the telephone during the Corona Virus Pandemic, this is no longer an issue. The relevant Registrars office will call you from the area of the country where the death occurred.
Pre Covid. If you did not live in the county where the death occurred, it was possible to give the details to any Registrar in England or Wales. This was known as registering by declaration. The Registrar to whom the declaration was given does not register the death, but would have sent the information to the relevant Registration Service. Once the death has been registered you would have received all necessary documents by post.