David Cameron has been accused by union leaders of being a "Bullingdon bully" after he vowed that the Conservative election manifesto would tighten the screw on strike laws in response to what he regards as Thursday's illegitimate mass walkout of up to 1 million public-sector workers.

Cameron attacked the low turnout thresholds in union strike ballots and challenged the validity of mandates to take industrial action derived from ballots conducted more than a year ago in some cases.

The prime minister said: "I think the time has come for setting a threshold. It is time to legislate and it will be in the Conservative manifesto."

In a sign of how the political battle may unfold, the education secretary, Michael Gove, will accuse the teaching unions of not standing up for education but for their pay and pensions.

On Newsnight on Thursday, Gove said teachers who were joining the strike were a minority.

He said: "The ballot which legitimates this strike is, I think, something like two years old and the turnout which validates that ballot was small.

"There are lots of people who, as members of trades unions, will respect the fact that it's a legally constituted ballot but I absolutely think that this strike is damaging. The truth is that there are a small group of people and they tend to be ideologically motivated and they are opposed to what we are doing."

He added that it was important that children could be protected from "politically motivated industrial action".

Cameron also rounded on Ed Miliband for neither supporting nor condemning the strikes, billed as some of the largest since the general strike of 1926. They cover teachers, civil servants, transport staff, firefighters and a range of local government staff protesting over real-terms pay cuts.

Dave Prentis, the leader of Unison, the largest public-sector union, also vented his frustration at Miliband's stance, saying: "It is time for Labour to make up its mind. Public-service workers are people who should be Labour's natural supporters and they deserve Labour's unashamed backing in return."

The unions in local government are seeking a pay rise worth £1 an hour. The unions claim ministers have in effect served notice that pay freezes in the public sector will continue until 2018, by which time the deficit is due to be eradicated.

Unite, Labour's largest financial backer, released a Survation opinion poll apparently showing that some of the union-bashing rhetoric of the 1970s and 80s no longer instantly chimes with the public mood. It showed that the public back the right to strike in this dispute by 61% to 31%, support a £1-an-hour increase in council workers' wages by 48% to 35%, and oppose public-sector real-terms pay cuts lasting to 2018 by 56% to 25%.

The poll did not ask voters if they support a change to the strike ballot threshold.

But Len McCluskey, the Unite general secretary, attacked the prime minister's plans to tighten the strike laws. "The whiff of hypocrisy coming from Cameron as he harps on about voting thresholds is overwhelming," he said. "Not a single member of his cabinet won over 50% of the vote in the 2010 election, with Cameron himself getting just 43% of the potential vote.

"If he practised what he preached then no Tory councillors would have been elected in the last 20 years and Londoners would have been spared the circus of Boris Johnson. So we'll take no lessons from the Bullingdon bully, who gives tax breaks to his City chums yet plots to deprive lowly waged workers of their right to fight poverty pay."

The Cabinet Office said: "This is a strike that will achieve nothing and benefit no one. The vast majority of dedicated public-sector workers did not vote for this week's strike action. We believe most people will come to work as usual, but rigorous contingency plans are in place to minimise the impact of action and ensure that key public services remain open."

The unions said up to a million people were expected to take part in the strikes.

The TUC general secretary, Frances O'Grady, said low-paid workers had borne the brunt of the financial crisis and the austerity drive and were still not benefiting from the fledgling recovery.

"The economy may be picking up, but having paid the price in pay freezes and below-inflation pay increases for several years there is to be no financial let-up for town hall employees and other public-sector workers," she said. "For them there are no shares to be had in the UK's economic recovery. Instead, several more years of penny-pinching and frugal living lie ahead."

Union leaders say more than a million workers were balloted before Thursday's industrial action. They plan more than 50 marches and rallies across England and Wales including protests that will end in a rally in Trafalgar Square. There will also be picket lines at schools, council offices, depots and fire stations.

But Tory MPs said strike action in schools had been supported in a ballot in 2012 by 22% of NUT members, and 33% of NASUWT members. Under British laws a single strike ballot can make successive rounds of industrial action lawful if the strikes can be shown to stem from the same dispute.

The Conservatives have been talking about revising strike laws for nearly a year, but have been forced to put their plans into the manifesto rather than legislation because of Liberal Democrat opposition.

The Tories are considering two strike threshold options. Under the first, backed by Johnson and Gove, a strike could only take place if it was supported by a majority of the entire membership, not just those who vote. Under the second, a minimum turnout of, for example, 60% would have to take part, regardless of how they voted.

The TUC published research on Wednesday showing that since the coalition took office, local government workers, NHS staff, teachers, firefighters, civil servants and other public servants were on average £2,245 worse off in real terms.

O'Grady said: "It won't have been an easy decision for hard-pressed public sector workers to vote to lose a day's pay this week, nor will they take delight in any disruption caused to the public."

"But if the government continues to hold down pay, our public services will struggle to hold onto and recruit skilled and dedicated staff. When that happens we all pay the price."