I've never had to obtain RN records for someone as late as this myself, but as your father would be next of kin that is someting that could be looked into.
I remember the struggle someone else had when trying to get the army papers for their father (similar circumstances) where the MoD wouldn't not accept the application as the death cert of the man could not be provided, because, like you, no one knew what happened to him. Typically, I don't remember what happened in the end though! Here's some info on obtaining Navy records:
http://www.mariners-l.co.uk/UKRNPersonnel.html