I'm so sorry for your loss. It's hard to lose a parent at the best of times and this is not the best of times. Glad your mum is ok.
My mother in law died last week. My husband made an appointment with the registrar in advance of having the info from the doctor (I think he lied and said he already had it when they asked). He was really worried that something would go wrong and he wouldn't be able to collect the form from the doctor in time (we live 2.5 hours away so important to coordinate visits).
Fortunately it has all gone well so far. We went down on Monday, stopped off at the GP surgery and then went to the registrar and then to the undertaker. I can't believe that they wouldn't relax the 5 day rule if necessary, especially if things were delayed a bit because of the involvement of the coroner.
As far as banks etc are concerned, as others have said they all do it differently. When my parents died I went into the banks and dealt with it all face to face. My husband is doing everything online for his mother and he has been surprised how smoothly it has all gone. He had power of attorney and had already consolidated things so that there were only two institutions involved, which made life easier.
He hasn't able to get a certified copy of her will but in the circumstances people seem to be working round that. If your father's will is lodged with a solicitor that shouldn't be an issue for you, as long as you are the executor. A complication I remember when my father died is that I was one of several executors and the solicitor wouldn't give me the original or a certified copy without written agreement from the others. It want a problem but it held things up a bit.
On the funeral, my mother in law had wanted a service at her church followed by a cremation which she wanted no one to attend. We're having to adapt that and instead will be having a short cremation with only immediate family ( no more than 5 of us) and hope that we will be able to follow this with a thanksgiving service at her church in a few months time.
Someone mentioned the 'Tell us once' service. I found it very helpful but it still isn't available everywhere. I think Brighton might not have it, for example.
The funeral director will sort out a notice in the local paper for you. Obviously you will want to agree the wording with them.
Good luck. I know we all grieve in different ways, but I found having all the practical things to focus on really helpful.