I'm sorry for your loss. 
My Dad died at the end of August so my Mum, my brother and I are still dealing with practical matters.
Once you feel able to, and when you know that the doctor has sent the medical form through with the cause of death, the first thing to do is to register the death, as you can't do much else without a death certificate. I rang the Register Office to see if I had to make an appointment and to my surprise I was able to do the whole thing over the phone immediately. This may be unusual, though.
Whether you do it face to face or over the phone, you need to have a lot of documentation to hand as there will be quite a few questions to answer. The links given above probably give a list. From memory, we needed my Dad's birth certificate for information on that about his parents, which I hadn't expected to have to provide. I think we may also have needed my parents' marriage certificate.
The Registrar asked us if we wanted to use the Tell Us Once service and when we said yes she did it immediately for us as part of the samel call. This is a really useful service available in England, Wales and Scotland. I believe NI has an equivalent service. Once the death is registered, this service automatically passes the details onto HMRC, the Department of Work and Pensions, the NHS, the DVLA, the local council and probably various other government and similar departments. That saves you a lot of letters and calls, and it all worked very smoothly for us.
It's a good idea to ask for at least one copy of the death certificate as you may have to send it off. There will be a small charge for that.
Once the death is registered, you can arrange the funeral. You should also notify your Mum's banks, pension providers (DWP will already be notified, see above) and any insurance companies. My parents' bank has a bereavement team who dealt with everything (again, over the phone, but in their branch, so that the staff there could verify ID first).
You should notify whoever is insuring the house and contents as soon as possible that the house is now empty so that cover isn't affected.
I hope this helps. It all seems overwhelming at a time when there is so much to process emotionally. It must be much worse if the death was sudden and unexpected. (My Dad's wasn't, really, as he was very elderly and in hospital.)
Very best wishes for the next few weeks.