Well, having worked in the cemetery business, this could be taken care of in numerous ways.
If the elderly person had arrangements made at a funeral home prior to passing, which many, many people do, the funeral home will transport the deceadant upon notification of death. They take possession of the body and arrange for the death certificate, etc.
If someone passes away at a hospital, unless there is a coroner's inquiry for toxicology and things like that, a funeral home chosen by the family or predetermined in the person's will, etc will take the deceadant for burial and/or cremation.
Early 60's IS young. Autopsies aren't routinely performed.
My mother in law was 67 and dropped and died on her kitchen floor one morning. It was obviously a massive coronary. The doctor did not order an autopsy and we didn't request one. Her mother, strangely enough, had died the same way.
Have you asked any of these questions of the funeral home? If you have papers showing that you are the legal executor of the estate, it seems you have a right to know who notified them, who took care of the arrangements, etc.
I have a little aunt in Oregon and I am her only relative. If she passed away today, God forbid!....she has already made arrangements, years ago, as to her burial wishes. She has the entire thing paid for already. The casket, the plot next to her husband, the headstone is already in place. All they need to do is place her where she wants to be and add the date of her passing. Those arrangements were made and paid for well before my uncle passed away.
It's not unusual for a funeral home to go into the person's home. It's not unusual for you not to receive a bill. It's not unusual for there not to be an autopsy.
I'm truly sorry for your loss. I certainly hope there wasn't any foul play although if it didn't seem like a passing from natural causes, I'm fairly certain the funeral home would have reported that as they need a physician to sign off on the death certificate.
Start with the funeral home, but if you really feel something was suspicious with her actual manner of death, you will need to speak to the authorities.
Again, I'm sorry for your loss.