highly unlikely. In most instances your insurance covers you (and any named drivers) to drive your car FC and, usually to cover other cars TP only. His insurance is probably either for him to drive TP only or maybe even not at all to drive other cars!
If you gave him the keys or left the keys unattended and knew that he had the car then you haven't got a chance to claim from him unless you sue him and even then he took it with your consent so you've got no chance of winning that one. If he stole the car then the insurer will want the crime reference number from when you reported it and, if you reported it after the incident I doubt the police will be interested and suspect the insurance company will not take you seriously. Obviously if you did this then he faces a criminal prosecution. If you did this after the fact knowing it was a way to get some resolution then this is fraud (obtaining money or services by deceit) and is also perverting the course of justice and wasting police time.
All things considered maybe your mate (if he really is a true and genuine friend) should reimburse you out of his own pocket. Good luck!