He suddenly changed his attitude and said he didn't want to go through insurance and almost begged me not to either.
I agreed.
You have to ask - why is he so keen not to go through insurance?
It may just be that he doesn't think he has a chance and doesn't want to lose his no claims - but if he's that desperate not to consult insurers, you have to question whether he actually has insurance, is a disqual driver, or has other things to hide...
I personally wouldn't agree to someone who's so desperate not to involve insurance companies, especially for an accident of such severity that someone was knocked off in a car-vs-motorcycle, to the extent even someone in the car got 'neck injuries'
The law's requirement for an RTC with no serious injuries is that you swap details. Which you have - so you've met that obligation. BUT I would still contact your local plod and get an incident raised. They will, when they raise this incident, do PNC checks etc on both vehicles for their logs - and if the chap indeed doesn't have insurance etc then it would be followed up.
The terms and conditions of your insurance will most likely say that you need to contact them and inform them of the collision, EVEN if you're not claiming. If the other party accepts liability (which it sounds like he won't) then you do have the option to deal directly with his insurers to recover any losses or injuries, however usually you must still tell your own insurance company that you're doing this.
Hope this helps.
I understand these things happen... but if liability is disputed questions will be raised such as did you do your blindspot checks, were you indicating, were your indicators working, was the M/Cyclist wearing visible clothing, etc...