It's broadly correct, though naturally skewed slightly (see Angus Calder's The Myth of the Blitz for the myriad ways aspects of "the myth" have been accentuated or downplayed by those across the political spectrum). Government policy up to 1940 had been for dispersed shelters rather than large public shelters, and for trenches or brick-built surface shelters rather than deep shelters; conveniently this was the cheaper option but also down to concerns over health risks and public order issues amongst large crowds, and fears over a 'deep shelter mentality' developing in which people would refuse to leave shelters, bringing cities to a standstill. The London Underground was closed to shelterers, both in accordance with policy and to prevent interference with transport. In response, according to Richard Overy's The Bombing War, "several thousand Londoners bought tickets for the Underground and stayed put in the stations and tunnels" on September 7th, the first night London was heavily bombed. The following night a large crowd gathered at Liverpool Street Station and refused to leave, the doors eventually being opened to them. At other stations "Minor confrontations occurred, orchestrated in some cases by Communist party activists, between crowds waiting to go below and Underground officials whose instructions were to lock the entrances once a raid began" (Geoffrey Field, "Nights Underground in Darkest London: The Blitz, 1940-1941", International Labor and Working-Class History, No. 62).

The government attempted to dissuade people, putting out a notice in newspapers on September 19th exhorting people "to refrain from using tube stations as air raid shelters except in case of urgent necessity", but made no serious attempts to lock down stations or evict people by force; on September 21st Home Secretary John Anderson informed Churchill that the use of the tube for shelter was a fait accompli and that only military assistance could prevent it, something that public opinion would not allow. By 25th September (prior to Morrison becoming Home Secretary) the use of tube stations as shelters was officially recognised.