Condolences @CalicoPusscat , I'm sorry for your loss.
With the new system, a medical examiner will need to review the medical certificate, and then notifies the registrar who then notifies you. You will need to then register the death within 5 days of the registrar's email. It was a quick process for us very recently - no hold ups with the ME. You might want to think about ordering extra copies of the death certificate beforehand as you may need to send off several copies at once once you start notifying organisations.
The registrar will give you details of the Tell Us Once service - you can do it online and it notifies HMRC, DWP, the passport office, DVLA, local council etc.
The funeral directors will be provided with paperwork by the registrar so will start what they need to do - they should organise everything for the funeral for you.
Once you have located your mum's bank / savings accounts details, go through the bereavement departments of each institution - they will be very helpful. Once you notify the bank, the account will be frozen to stop payments going out, but payments can still go in. They will also settle certain bills, such as the funeral directors.
If your mum had a house, then you'll need to let the insurance company know, and also the gas / electric / water / phone providers - again, use the bereavement departments, they will explain the best way forward.
You'll also need to notify any private pension providers (the Tell Us Once service will do some public sector schemes).
Hope the above helps - take care of yourself.