I can give a little insight, we lost my dad on NYE, passed at home where he wanted to be, in his own bed with me, my mum and my sibling all holding him and reassuring him.
He passed early hours of the morning, the hospice were involved in his care in the last few months and were very much on hand the last couple of days. We knew it death was imminent and a RESPECT form/DNR had been completed, they arranged a team of carers to come by who done some personal care and made him more comfortable.
After he passed we called the hospice again who came out as soon as someone was available as it was about 4am, they came about 8am and confirmed death. Weirdly time of death is recorded as that time, not the time we know he passed.
The hospice asked if we knew what funeral home we wanted to use (which we did), and I think another team of carers came out and dressed him for us at some point that morning.
2 people associated with the funeral home came out to collect him but I can't remember what time, it was quite a lot later that day I think, I know it was long enough that my mum got back into bed and cuddled dad while she tried to have a little nap. These 2 took inventory of the clothing and possessions he had with him and gave us a receipt for these, then transported him out of the house and into their van.
It then took a whole week before his body was properly released to the funeral home in terms of them being able to start embalming processes etc because someone needed to confirm cause of death before we could register the death, this is apparently normally much quicker because the person's GP or hospital/hospice doctors can sign off, but because my dad was cared for by the hospice nurses and never actually saw their doctors, and hadnt seen his own GP for a couple of years, his ENT doctor had to go out, view him and sign off.
Once that bit was done things started moving quite fast and the funeral directors could start their processes once we had registered the death. By this point we had already had our appointment with the funeral directors so they knew what we/dad wanted
Edit: we did call 101 when he passed as we were told to by the hospice, as they thought perhaps a duty doctor would be able to come out to pronounce, but the hospice nurse arrived before they did (I'm assuming the hospice updated the duty doctor as I don't remember them getting back in contact with us