Why were British laypeople more ready to challenge the government starting in the 1960s?

I don't think there's any evidence to justify the idea that before the 1960s the British people were less prone to challenging the government. Britain has seen widespread public anger with status quo and government at various times for various reasons.

In 1819, in the shadow of the Napoleonic Wars and French Revolution, British cavalry charged into unarmed civilians protesting to demand representation in parliament. This is called the Peterloo Massacre. Prior to this, during 1811-16 Luddites had been destroying so much factory machinery that the army was deployed to Nottinghamshire to return the region to a lawful state.

The 'Battle of George Square' in 1919 saw Glaswegian workers on strike brawling with police, resulting in the deployment of the army to restore order. Come 1926 the General Council of the Trade Union Congress called a general strike which lasted nine days and involved 1.7 million workers.

During this era following the Russian revolution of 1917 the British government was concerned that a large strike would be the start of a communist revolution.

However, while it is true that the 1960s was a unique time which did provoke anti-establishment feeling and protest, it is also untrue to suggest that this was thus the first time the British people were disobedient or revolutionary.

There was political turmoil worldwide during this era, so anything going on in Britain has to be understood as being part of that trend.

In America and the Soviet Union old certainties were being tested. The Civil Rights movement in America, the Counter-Culture, and the Vietnam War, all eroded public trust. This combined with the rise of a post-modernist rejection of objective truth did a lot to upend previously unifying beliefs. In the USSR the death of Stalin, the Hungarian Revolution, and the Prague Spring together did something similar. The Sino-Soviet split also occurred between 1956-66, in which the USSR and PRC denounced each other with increasing intensity. All this was against the backdrop of a mushroom cloud, as America and Russia had been doing more and more nuclear testing of increasingly apocalyptic weapons.

By the 1960s the world was seeing a total collapse of certainty, and this confluence of ideological criticism touched on every aspect of life and authority.

You are basically asking why "the 1960s" happened. They were a phenomenon across the western world and sometimes beyond. This is a large question but reasons may include:

The post World War II 'baby boom' meant that there were more teenagers and people in their early twenties in the population so their influence (and buying power) tended to be felt more.

Greater prosperity; the young could more easily afford drugs, rock and roll records and also books, magazines, foreign travel and television that allowed them to experience a wide range of ideas and ways of life.

More effective and more widely available Contraception allowed a change in sexual behaviour that probably caused a more general questioning of family values and old ideas of respectability.

In Britain the 1960s were mostly a time of peace. The solidarity, discipline and austerity that the majority had mostly accepted during and immediately after World War II no longer seemed so necessary.

Also, in the 1960s Britain was rapidly losing most of what was left of its Empire and was economically being overtaken by other European countries like Germany and France. Consequently trust in institutions like the Civil Service, Parliament, Government and major companies was declining as they no longer seemed so obviously better at running things than their foreign counterparts.