It was a propaganda effort at the time, 1940, to get American public opinion on the side of the British. It was not a sure thing that America would not just sit out the war. Joseph Kennedy, as ambassador, said that England was done.
Firearms laws were not particularly stricter in the UK than in the U.S. at the time. Perhaps fewer people owned guns, but fewer had the economic ability or perceived need to. Hunting was not a sport for everyman in England and many of the urban poor, likely to buy a gun for protection, just could not easily afford one. The upper class liked it this way as they historically feared ending up like the French and later Russian aristocrats did in case of social revolution. (You could be sentenced to death for urging dissolution of the monarchy on through virtually the modern era.)
It was like collecting tin cans for the war effort. It makes people feel involved and makes for good propaganda, but does not really have much impact.
The British Army was indeed short of weapons in 1940, they left many of theirs in France when they evacuated their troops.
To be in the Home Guard, established after Dunkirk, you had to have fired three rounds from a rifle or shotgun. That was the training standard. They organized training days that amounted to range days.
Even in today's United States, relatively well armed, probably two out of three people do not own a firearm. Thus if we tried to arm everyone to fight the Martians, we too would have a huge deficit in firearms.
Fully automatic weapons were legal for ordinary folks until 1968 in England. At that point they had to be converted to semiauto only. Military style semi autos were legal until the 1980s, all long after WW2.
Many rifles, shotguns, and yes even handguns (blackpowder, historical, or Buntline style revolvers) can still be had, albeit with more paperwork and problems.
Those old ads you see were the work of British propagandists/intelligence agents in the United States. It was...well perhaps scam is too strong a word, but it was not the historical lesson that it is sometimes made out to be.
