Unless you have religious reasons for doing otherwise, take your time. If someone dies at home, by all means call a funeral director and ask them to collect the body but be aware that you can have them transferred to another funeral director for a nominal charge before any paperwork is signed.
This also applies if the person has already been collected because they died in a nursing home. If the person died in a hospital there may be no rush – they can stay in the mortuary until you’ve chosen a funeral director you’re happy with. If the hospital does not have a mortuary, a nominated funeral director will look after them until you arrange for a transfer.
By all means call family and friends to tell them that death has occurred, but don’t feel that you need to tell them the place and time of the funeral in the same call. Unless the coroner is involved you must register the death within 5 days.