After the industrial revolution, Britain became the factory of the world, ranking first in the world economy and national strength. The state can and must provide it.
Because although the industrial revolution has enriched the content of improving people's living standards, the social problems brought about by it have become more and more acute, and sweatshops bear the brunt. As the name implies, workers under the factory system are not better off but more oppressed, which directly leads to the emergence and increase of social class contradictions. In order to alleviate contradictions, social security legislation, such as welfare policy, has been implemented.
Therefore, the welfare policy of capitalist countries began when the economic capacity allowed and the society itself needed it, and it gradually became the public policy of all capitalist countries. Especially after Roosevelt's New Deal, it has become an important means for capitalist countries to cope with economic crisis and ease social contradictions.
However, when you say complete welfare from cradle to grave, you should mean 1948. At that time, after the end of World War II, it was necessary for Britain to improve its welfare system in order to stabilize society.