In the UK any airgun that can deliver over 12 foot pounds of energy at the muzzle requires an FAC "firearms" permit. Your average pump up pellet gun will usually deliver around 8-9 foot pounds and a daisy red ryder BB gun is less than 1.5 foot pounds of energy. So at 7 yrs old I am guessing he is using a red ryder and 1.5 foot pounds of energy is in no way comparable to a firearm.
Here in IL USA, they require you to have a firearms card to purchase any airgun that is .22 cal or larger but they still aren't considered firearms and aren't treated like firearms, don't need to be registered etc.
IL is not a very friendly firearms state. The only state in the US that does not issue concealed carry permits. But the supreme court ruled it unconstitutional so at some point in the near future we will be able to get CC permits.