This is the budget that Downing Street hoped would demonstrate to its new voters that they were right to place their trust in the Conservatives. Instead, it has become the coronavirus budget, with No 10’s message to voters in the north and the Midlands half-postponed while the Treasury tries to protect the economy.

But the moment to reassure those new voters will come eventually. Until now, the government has signalled that its view of how to do this is through “levelling up” the country – improving infrastructure, transport links and high streets outside of London and the south. We have heard about Northern Powerhouse Rail, better bus routes, and a trans-Pennine tunnel. We are told that the route to lock in the new Conservative vote is through infrastructure.

Among voters who switched from Labour to Conservatives, the top priority is NHS – and improvements are an expectation

However, research I have conducted since the election with these new Tory voters suggests this approach is overstated and runs the risk of misreading the group. For all the talk of new infrastructure projects, voters will be judging the government on something much more totemic, more closely linked to their values – the National Health Service.

The view that the Conservatives’ fortunes depend on the extent to which they can deliver is accurate. The voters I have spoken to in focus groups in Tory-gained seats are open to voting Labour again. They say Boris Johnson will need “to get things done”, “to keep his word”, if he wants to keep their vote. But when saying this, people are usually thinking about the NHS – and the expectation that there will be real and tangible improvements over the next four years.

This is borne out in the polling we conducted. Among the people who switched from Labour to the Conservatives, their top priority for the government is “investing in the NHS”. Of a list of the promises the Conservatives made at the election, “50,000 more nurses” is deemed most important. And when people were asked explicitly how they will judge the Conservative party’s record at the next election, 62% say it will be based on whether the NHS has improved. This is a higher proportion than those who say the most important criterion is “a clean break from the EU” or “the performance of the economy”. Strikingly “whether my local area has improved” is top for only 9%, and “whether transport infrastructure has improved” for 3%. Worryingly for the Conservatives, this is more than a mere wish: improvements to the NHS are an expectation. Of all the things they believe the government will actually do, investing in the NHS is the first.

What do people mean when they say they want improvements to the NHS? They mean more doctors, more nurses, more beds – improvement locally as well as nationally. They mean they want to see the new nurse bursary promised by the Conservatives, a pledge that has achieved near-universal cut-through. It is not only more money; voters also want to see waste reduced. They talk of back-office functions, overpriced medicines and managerial culture diverting resources away from the frontline.

People are not asking for the moon. The public are cleverer than a lot of pundits give them credit for, and know that a government cannot improve everything overnight. Rather, they want to see movement in the right direction. A bigger risk for the government is exaggeration. Voters feel they have been “done over” by party promises before. If the government inflates figures to illustrate results, it could backfire hugely.

The NHS is not the only game in town. Immigration is also important – these voters want to see a reduction in people entering the country as well as controls on who can come in. Though studies show immigration receding as an issue across the public as a whole, it is still one of the first things to come up among voters who left Labour last year.

Even a starved NHS is still our best defence against the coronavirus | Polly Toynbee Read more

There are other things they want to see. Better parking, business rates, better jobs, investment in rail, less congestion on the roads. All of these things feature and are not to be dismissed. But politics is about prioritisation. And overall they come nowhere close to the centrality of the health service. That is going to be the benchmark on which this government is judged. At one of my focus groups, a colleague asked why the NHS was so important. The reply? “It’s in our blood.” You don’t hear that about a new bus route.

Demonstrating progress on the health service will be crucial to the Conservatives’ success. Boris Johnson would do well to put it at the forefront of his public communications over the next few years.

And in many ways, this budget is going to give voters what they want. Because of the coronavirus, it is likely to be quite heavily focused on the NHS and its operational readiness. But with the same voters so sensitive to the state of the health service, the political risks coronavirus brings are immense – especially given the pressure it may have on hospitals and intensive care units.

Transport, infrastructure, high streets – for all we hear about them they pale in comparison to the National Health Service and, also, immigration. That is unsurprising. Look back at the 2016 referendum campaign, where immigration was the top reason for people voting leave and where the NHS – right down to the red bus – was key to the leave campaign’s victory.

The leading lights of Vote Leave are now in government. And, when it comes to how they are judged, they will have to reap what they sowed.

• James Johnson is a former Downing Street pollster who worked under Theresa May and now runs JL Partners