Not cheery stuff, but possible useful one day… If a sole practitioner has died, temporary emergency authorisation from the SRA is required. The SRA need to be notified within 7 days and the application made on form FA7 with 28 days of the date of death.
SRA guidance is here – http://www.sra.org.uk/solicitors/firm-based-authorisation/tea-sole-practitioner.page