Do we have to use the funeral director appointed by the Coroner?

No, you don’t.  If it was yours, or the deceased person’s wish, to use a specific funeral director then you are free to make that choice. A coroner becomes involved if a death has been sudden or unexpected, and the deceased has to be taken to a hospital for a post mortem examination. You do not have to use the services of this funeral director for the funeral arrangements. The funeral director who took care of your loved one via the Police or Coroner is appointed by the Coroner for this role only.