Im assuming youre asking in regards to your no insurance charge. The HTA grants you six days to register the vehicle upon becoming owner, so I would say registration is not required to show ownership.
That being said, in your particular scenario I think youre going to have an uphill battle convincing everyone that you had lawfully purchased the vehicle. The fact that your friend hadnt registered it or properly documented the initial sale takes away a lot of credibility from your claim. I think it's certainly sufficient for your insurance company to dispute the matter.