Your county Coroner is totally independent of hospitals or the police. A coroner will call upon the services of a number of pathologists to perform postmortems on cases his office is reviewing. As you can imagine there will be many (more than you think) to be performed and it may take some time for the PM to be done and the notes to be written up and sent to the Coroner. There may also be toxicology tests to be done - these will be carried out at the Coroners instruction at a licensed lab - again the results have to be submitted back to the Coroner. The Coroner may also request a report from your dad's doctor of his medical history.
All this can take some time to put into place - it does not mean that you will have to wait weeks or months for the coroner to decide that you may go ahead with Dad's funeral - he/she just have to be certain that they have gathered all the facts.
Do phone the Coroner's office and ask to speak to his/her Personal Assistant, ask them if you can have the name of the Coroner's officer handling your case. I've met many and I can tell you that they are very professional, kind and will, where they can, give you a great deal of support and advice.
Sending you best wishes at this sad time.