It is well established that, if you weigh up the life-years gained through cycling (due to increased physical activity) versus the life years lost (due to injury), the health benefits of cycling far outweigh the risks involved. One widely quoted figure for the UK, acknowledged by Government, puts the benefit:disbenefit ratio for the UK at 20:1. Other ratios for other countries are higher still. (N.B. some of the academic references reduce the ratio by including the negative effects of pollution - however that's obviously irrelevant to the helmet debate. If you remove the pollution effect, the other references all come out with ratios above 20:1). But let's take 20:1 for the sake of argument.
From this, recent research shows that, if you tell people to wear helmets (whether by law or simply through promotion campaigns) and this reduces cycle use by more than 1 unit of cycling (e.g. one cyclist, or one km cycled) for every 20 who continue, this is absolutely guaranteed to shorten more lives than helmets could possibly save - even if they were 100% effective at preventing ALL cycling injuries (i.e. leg, arm, shoulder injuries as well as head injuries) for the remaining cyclists. That maximum threshold, beyond which you would be doing more harm than good, then drops further still - down to c2% - once you take account of the proportion of cycling injuries which are non-head injuries. And this is still assuming that helmets are 100% effective at preventing head injuries.
In fact, the evidence on the effectiveness of helmets has become increasingly sceptical over time. A recent literature review by Rune Elvik, an internationally recognised authority on road safety, found that the estimates of helmet effectiveness have progressively decreased over time, with the most recent studies showing no net benefit. In this same report he documents evidence that helmets increase the risk of neck injuries. In a separate report, Elvik has also found that helmet-wearers suffer 14% more injuries per mile travelled than non-wearers. The reasons for this are unclear, however there is good evidence that (at least some) cyclists ride less cautiously when wearing helmets, and that drivers leave less space when overtaking cyclists with helmets than those without.
The only clearly documented effect of enforced helmet laws (e.g. in Australia, New Zealand or parts of Canada) is to substantially reduce cycle use, typically by about a third. Reductions in cyclists' head injury have been similar to the reductions in cycle use, suggesting no reduction in risk for the remaining cyclists, and in some cases this appears to have worsened. In addition to the possible explanations in the para above, this may also be becuase reductions in cycle use undermine the "safety in numbers" effect for the cyclists who remain - see see
www.ctc.org.uk/safetyinnumbers. A clear relationship has been shown between cycle use and cycle safety - cycling is safer in places where cycle use is high (e.g. the Netherlands or Denmark - or within Britain, in Cambridge or York). Telling people to wear helmets, instead of creating safe cycling conditions, is contrary to the aims of encouraging more, as well as safer, cycling.
From this, I hope it is clear that the effectiveness or otherwise of helmets is not the main point. As explained above, even if helmets were 100% effective, you would still be doing more harm than good if you deter more than c2% of cycle use by telling people to wear them. That's because the risks of cycling are not especially high, and the health benefits are SO much greater. You are about as unlikely to be killed in a mile of cycling as a mile of walking - do we also need walking helmets? - no, of course not! The idea that you need helmets to cycle is both a symptom of our massively exaggerated concern about the "dangers" of cycling, which results in such pitifully low cycle use in Britain.
In short, if we want to maximise the health, environmental and other benefits of cycling, we need to focus on creating safe conditions, and thus increasing cycle use. Resorting to helmets simply tackles the symptoms of the problem, not the causes, and thus deters people from cycling. This is pretty much guaranteed to shorten more lives than it could possibly save. Faced with both an obesity crisis and a climate crisis, the last thing we should be doing is driving people into increasingly car-dependent, obesogenic lifestyles.