I am so sorry you are going through this. When my mother was dying I was told to keep speaking to her. The hearing is the last sense to go. Emotionally I don’t know how to help you but on the practical side there are thing to do, but not to rush.
After her death the hospital will make you an appointment for you to get the death certificate. I am assuming there is no home to sell, but there are people to be notified. The hospital will issue youwith a folder which will guide you. Nowadays the DWP and tax offices are notified under the “Tell us Once Scheme” . You then have to register the death, you will be told where to go, and you will be issued with a certificate to give to,the undertakers. You can arrange the funeral or cremation before you get the paperwork.
Check whether your mum had private pensions or Spouse pension and if she had any life policies. Any bank , post office or Building Society will need to be notified. Also she may had had Utility shares so the Share Registrar will need to be informed. If so I would get at least 4 death certificates as people are quite bad a sending them back.
You can , if you wish, ask the undertaker to take your mum from the hospital to the Chapel of rest. In my case we decided not to have the funeral come to our home. We all met at the Crematorium. My Stepfather had little money, so we had no funeral cars and a basic but very nice funeral. It was £3200.
Things that you will need to tell the undertaker, are whethe your Mum will be in an open casket at the Chapel of Restl if so you may want to choose her favourite clothes. If not then a shroud is provided by the undertaker. You will need to think about music, and who will officiate.
Undertakers have their own list of lay and religious people, or you can choose someone yourself. If it is a cremation you will also be ablee to select an Urn for the Ashes if you wish.
We asked people back to our house for after the funeral but of course pubs or local halls will also do this.