Possibly, although like I mentioned in the other thread, car insurance is generally a lot more expensive in Canada than it is in the US, so if cheaper auto insurance is an effect of UHC it’s a fairly minor one. Although, granted, part of that is that most drivers in the US really aren’t carrying enough insurance to cover an accident that results in multiple serious injuries. So we don’t necessarily see the full effects of health spending from auto accidents in our auto insurance rates, because once the auto policy limits are exhausted the costs usually get shifted back onto the healthcare system in one way or another.
Plus, I’m only (somewhat) familiar with the situation in Canada; maybe other UHC countries do actually dispense with bodily injury coverage on car insurance like you’re thinking. I think in general, though, the attitude is that healthcare is a basic human right but driving a car isn’t, so countries try to make it so only people who drive cars shoulder the costs of traffic accidents.