I'm going to be honest, actual statistical impact is going to be hard to find - how are we going to control for the fact that "the kind of parent who tends not to bother taking the coats off" is also likely to be "the kind of parent who drives a bit less carefully"? It's the same kind of confounding factor that makes breastfeeding studies really hard to do.
I suppose you could try to find some area of the world where most people currently do not bother about this kind of thing (Japan, LOL?) and then do a big campaign and see if child deaths in cars go down, but then child deaths in cars are already rare enough that it's still going to hard to find impact.
I mean, people should use seat belts and children should use car seats, these things make very big differences to the risk of dying in a car. Once you get beyond that point and get into the whole realm of worrying about this little detail and that little detail... I'm not saying there is no impact but it's likely to be very small and we are probably well into the realm of diminishing returns at this point.
I've said this before on these threads: that with most people using car seats correctly already, society (if it wants to push down death rates still further) should be focusing more of its energy on the more holistic stuff, like vehicle speeds, the number of cars on the roads, raising the minimum driving age, hvaing proper testing for drivers over a certain age, and making sure there are alternatives to driving for most people so that heavily sleep deprived parents don't wind up driving their kids about while cross-eyed with exhaustion because they have literally no alternative. These steps are unpopular in the UK, however.
For what it's worth, yes, I do use car and bike seats correctly and remove coats that are thicker than a knitted layer or hoodie/thin coat.