I'm sorry to hear about your Dad.
Is this sudden so they have had no time to start to do any of the admin?
I'd not assume your parents have done nothing unless you know that for definite.
There are things that you and your Mum can't to until (sorry!) you have a death certificate. Once you do, you can send it to the various places who need it, so that any money in his name is transferred to her.
I was involved in this when my Dad died a few years ago.
You should ideally appoint a solicitor who can advise you on what to do and who to contact, if neither you nor your Mum know.
Your Mum will have to do everything, in her name, as she's next of kin. You can advise her but she will have to do the phoning and writing- you can't do that for her if she has 'capacity'.
There isn't any real urgency to be honest. As long as she has access to money such as her own account or a joint account, you don't need to do anything at the moment.
TBH I'd spend your time with your Dad and supporting him, rather than spending too much time and energy on paperwork just now.