Everyone was issued a gas mask by the outbreak of war, and government campaigns urged people to carry them at all times; most did to start with but as the feared air raids failed to materialise compliance rapidly declined. A survey on Westminster Bridge in November 1939 recorded 24 per cent of men and 39 per cent of women carrying masks), Robert Mackay in Half the Battle gives a figure of 5 or 6 per cent of Londoners carrying masks in February 1940 from Mass Observation. Once German raids started the figures rose, but only as far as 30 per cent in June 1940, and fluctuated with the weight of attacks; with the lack of poison gas it wasn't a priority for most, in a 1942 comedy film prospective German spies are instructed "To be perfectly English you must always remember to forget to carry your gas mask". They could be useful, though; Juliet Gardiner in Wartime quotes Angela Culme-Seymour: "I made sure we never forgot to take our gas mask when we went into the shelter during a raid on Kensington where we were living. It made an ideal potty for the children."

The blackout was a slightly different matter, and perhaps a better parallel for the current situation; failure to carry your gas mask wasn't illegal, but breaking the blackout was with almost a million prosecutions by the end of the war. See my reply in a previous question here “You’re the sort of person who would have kept their lights on during the Blitz”. Was there a large amount of truth to this claim or not? for a bit more, and for general comparisons there's also It’s incorrect to say there was no resistance to safety measures around the Blitz at Full Fact and [The psychology of protecting the UK public against external threat: COVID-19 and the Blitz compared](https://doi.org/10.1016/S2215-0366(20)30342-4) in The Lancet.