Of the three car accidents I've ever had (none caused by me) one was a cyclist who ran into the side of my car and landed on the bonnet (totally his fault) and the other was an uninsured, banned driver who also drove into the side of my car at a junction.
Both times I've had to foot the bill. The idiot cyclist had come up onto the pavement, against the flow of traffic, and ran into my car as I waited to join a queue of traffic. The banned driver drove straight into my driver's door as I waited to turn right into a road where he should have been giving way.
Yes. Cyclists should have at the least third party insurance as they can end up on your windscreen (with their bike scratching your bonnet) - though I don't think anything can really be done if they cause an accident like in the OP and carry on.
As for uninsured drivers, the bloke who hit me, he had several DD offences to his name, swore blind to me he was insured (and to be fair did give me his correct name, address etc) only for his wife to ring me just after I'd reported it to my insurance to say they'd just realised his insurance had run out. It hadn't - he'd never been insured as he was a banned driver.
He was taken to court and fined £300 for driving without insurance, and then said he couldn't pay anything to my insurance company as he was on benefits, couldn't work and had already paid his court fine. Funny that, he was on his way to work when he hit me. Wanker.
I've still got his phone number and address. The fact he's a convicted arsonist and burglar are the only reasons I haven't sent some burly men after him!