Long wait for death certificate - what to do?

My MIL passed away on Sunday after a brief infection that ended in sepsis. The decision was taken to withdraw care and put her in palliative care and she stayed in hospital until she passed from the care home she resided in.

We have been told that it will be 7-10 working days for the hospital (NHS) to send the MCCD (death certificate) to the local authority. Nottingham City Council, if that helps.

I think that seems a really long time, as it could be up to two weeks. I don’t think any part of her death hits the criteria for coroners office involvement.

What can we start to do in the meantime? As we cannot really inform anyone of the death until we have the MCCD, so obviously her care/benefits accrued in that time will have to be paid back (which is fine of course!) but it’ll be a larger outlay initially for my partner who is already looking at 5k for a funeral.

This is a first time major bereavement, so I guess my questions are:

1. Is this a usual timeframe?
2. What can we be getting on with in the meantime? My husband is very practical and wants to get the ball rolling to sort things.
3. What will happen to her? Will she be okay for so long if it is two weeks? Then obviously the wait for the funeral?

Thank you so much!