90% of the delay is waiting for the medical person who verified the death to complete the paperwork and send the medical certificate for cause of death (MCCD) to the registrar. Without that you cannot register the death and get the "green form"- certificate for burial or cremation. We need that green form for the crematoria/cemetery/vicar 48 hours before the date of the funeral.
Some GPs/hospital doctors are faster than others. I had one when the GP verified the death and went on holiday for 8 days and no one else could fill in the form. The family then had to wait another four days for a registrars appointment.
Some examples (based on lots of individual funerals):
Someone passes on the 20/07, doctor is a hospital doctor, bereavement office let family know as soon as the MCCD sent to registrar on 22/07. Family get registrar appointment for 24/07, come straight to me with the green form. They want it as soon as possible. I phone our booking line and the only free spot we have is the 2nd August. I phone the crematoria locally and they do have a space. I talk to the family. I ask if they want an order of service (yes), if they want a celebrant (yes) and if family members need time to organise time off work (yes). If the answers to those had been no then they could have had the 2nd August.
If they had wanted a direct cremation then it could have been even earlier as we would have had a complete set of paperwork.
I understand that other faiths bury their deceased and they do it before all the paperwork is complete with the understanding the deceased can be exhumed if needed.
A whole new system is coming in in September which will make the process even longer, but at least removes the GP more from the equation and should make funerals £82 cheaper as we won't have to pay doctors on your behalf for the paperwork.