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I've been pondering this question in my head and I can't work it out the proper answer. Google is no help either.

A person (I'll call her Jane) owns a car which has comprehensive cover, and she is the sole insured person for the vehicle. While driving, Jane and her car are involved in an accident in which she is killed and her car is a total loss.

The question is: Does the insurance company pay out on the total loss of the car in this situation? And who and how would someone initiated such a claim? I assume any such payout would be made to Jane's estate and distributed accordingly.

In a situation such as this, the payout from the insurer would go to the estate. The fact that the vehicle was a total loss really makes no difference, as its monetary value is treated no differently from how a motor vehicle would be treated as part of any estate.
Normally, the lawyer for the estate would deal with the insurer.