There are already good answers (and even accepted one), but I want to add some information about statistics and genetics that would influence the reactions.
First, considering that you have about 5 new magic users per month. For 18 years, that make 1080 magic users in total (0.000015 % of the population).
The reaction of the government and/or the people vary greatly with the distribution of those magic users. And it comes down to how they came to have that gene.
Are those 5 children per month already sharing a genetical origin? Meaning once upon a time there was a wizard or a witch who had plenty of children, and some centuries later, those who inherited the traits start to develop magical properties?
This would concentrate the magic users in some culture. Say, e.g., that the witch was Polish. Most of her descendants would be located in Poland and around, probably a substantial part in Russia, Germany and the USA. Maybe a few more in Western Europe.
It depends on when the said witch was living, but as an example, that could possibly make 700-900 magic users in the region where she used to live in Poland. 50-100 in both Germany and Russia. 40-140 in the USA. Some 20-30 in the UK, Irland and France, add a few others in Baltic, Slavic and Scandinavians countries and there you are.
Of course she could have been travelling. Or be long ago where her genetic pool have travelled more (e.g. Huns, Mongols, Ancient Chinese, etc.). That would spread the occurences, but probably still along the commercial ways of the time.
Assuming a concentration in Poland, you have to consider that 6-15 are already dead: stillborn, death in infancy, etc. Which in that case would be negligible.
In that example, the government of Poland would probably learn about the strange situation and by discussions with EU, and NATO, most of the Western World countries would know about it. The high concentration, coupled with the occurence of cases elsewhere would directly point out to the genetic origin of the effect, and they could predict (to some extend) the likelihood of the new-born children. A few missing out in bastard lines. It would be pretty hard to abduct the children without being noticed, because of the high concentration. Furthermore it would be hard to bloc that information to spread that in some region a lot of children are strange. Nevertheless, I expect, the governments would try to impose a martial limit on the inhabitants. Probably closing the whole region off. The few cases that could be seen outside would probably be sent to study institutes.
Now instead, we could suppose that Magic (itself), or some other way creates mutations in the genes of babies-to-born. This could be equally distributed. If that were the case, all countries would be affected, proportionally to its inhabitants.
That would make a different count: about 150 born in India, 200 in China, 150 in Africa, 75 in Europe, 40 in the USA, etc. Considering the infant mortality rate of some of those countries, it is likely that already 100-200 are dead by now. Ignoring the children under 3 years old, which might be slightly unnoticed, we are down to, say 800 children in total. Most of them being in India, China, South-East Asia and Africa (around 510).
It is likely, in such a case that some regions would consider the children devil and basically lynch them. Countries like China, having sufficiently large amount of users, would embrigade them into some "schools" to see the military/economic potential, without caring about the parents too much. The USA and Europe might just set plans to identify and abduct them. The rate being low enough to get unnoticed in those countries. Secrecy from the main public might be kept to some extend.
Now would the genetic factor be identified? It is not so likely. Indeed in countries with stronger states, the children could be found from a relatively early age and kept separated/isolated for studies. In other countries, you might see some teen pregnancies, but there, the tracking is fuzzier.
The mode of acquisition of the "magic gene" greatly influence the reaction, the spread of information and the number of users.