Err, no, they are worried about a lot of Eastern block military surplus automatic weapons that were deactivated to older standards and ARE turning up reactivated. Yes anyone who can work a CNC machine can probably make a gun - but starting with half the work done for you makes it so much easier. This is not paranoia, it is in reaction to a Europe wide problem. Perhaps if the Police and Home Office publicised the issue a little better instead of keeping schtum about a possilble route to illegal 'Section 5' firearms acquisition the 'shooting community' would STFU.
If you or your kids live in a metropolitan city then there is the vanishing small, but still real, risk of being caught it a 'drive by' because your local drugs gang think its cool to own reactivated Skorpions. Unfortunately it is probalby extremely difficult to frame a law that makes an overly subtle differentiotion between e.g. Boer war and cold war era weapons.
Yet criminals also find it worth their time manufacturing their own, because the supplies of "weakly deactivated" firearms in the UK have dried up, no legitimate UK dealer or auction house will touch anything that doesn't bear a certificate of deactivation from the London or Birmingham Proof Houses and Border Force will take a dim view on people trying to import such.
It is very sensible to make it inconvenient for criminals to acquire firearms. But once you have reached that point (and we are well past it), it can only be addressed through enforcement.
Even a total ban on private ownership of firearms would not prevent criminals smuggling or manufacturing their own - lathes aren't licensed.
Additionally, whilst there is some concern about "automatics", firearms crime committed with automatic weapons is more or less zero. 50% of all firearms crime is committed with pistols (that ban in 1997 worked then), and a chunk of the rest is with "imitation firearms" and then perennial favourite - the sawn-off shotgun. There are a few notable cases - Ellis/Shakespeare being gunned down with a MAC-10 in 2003, but in most cases it seems the (non-)availability of ammunition to feed a full-auto along with their relative rarity makes them at most a status symbol rather than a practical weapon.