OP I work for a solicitors firm and specialise in probate (not a solicitor though) so I deal with this every day. The most important thing is not to rush into doing anything, take your time to come to terms with it first so you don’t make any rash decisions.
Once you’ve got the death certificate you should choose a funeral director, call around and ask for a rough price first, people never shop around and consequently some people end up paying over the odds for the same service. Take time to discuss with your father what he wants, a good undertaker won’t push you.
You don’t have to go into banks to register the death and there’s no time limit to tell them either. You can call them and ask for the bereavement department and ask if probate is required then post a copy death certificate to them or drop one into branch at your convenience, they may try to sell you an appointment with their probate services, you’re not obliged to give them information other than the fact she’s died and your contact details.
If you do need probate it’s actually very straight forward for most people, so you should be able to do it yourself if it’s under the inheritance tax threshold. You only need probate if a bank or other investment tells you they won’t release the money without a grant of probate. For the house you will have to find out how it is owned, so if you know which solicitors they used when they bought it give them a call and ask, they will tell you if you need to do anything but usually there is no rush.
Ask your dad if your mum wrote a will, in my experience it is almost unheard of for one member of a married couple to write a will and the other not to know where it is and what’s in it.