The lockdown could be eased further next month even if the troubled “test and trace” plan to track new infections is hit by further delays, No 10 says.

Both the tracking scheme and the launch of the smartphone app alongside it have been put back until “the end of the month” at the earliest – a two-week delay – with phase two of restriction-easing due to start on 1 June.

But Boris Johnson’s spokesperson said phase two would not be delayed even if testing and tracing was not ready – and insisted there was no “explicit” link.

The plan to push ahead with phase two is certain to alarm MPs and others who criticised the first cautious lifting of some restrictions last week, before the contact tracing scheme is up-and-running.

The rollout was due to start this week but will now only get underway, at best, as phase two of lockdown-easing – including schools reopening for some pupils – is due to begin on 1 June.

The spokesperson said the key factors were the number of coronavirus cases and the ‘R’ reproduction rate, not the tracing programme. “We will only move to step two if we are satisfied it is safe to do so,” they added.

The comment came as No 10 also:

* Refused to say whether schools or councils would be punished if they refused to cooperate with the government’s school-reopening plans.

* Revealed that 38 per cent of care homes in England – a total of 5,889 – had suffered a suspected or confirmed Covid-19 outbreak.

* Admitted the promised app might take longer to introduce than a physical tracing programme, saying: “It’s certainly possible that may happen.”

* Defended the immigration health surcharge paid by foreign NHS workers – after The Independent revealed a promised review never took place – arguing it “broadly reflects the cost of providing treatment”.

The test and trace programme has been hit by recruitment and training problems, on top of widespread criticism that ministers have bungled how it will operate.

A leader in disease control lockdown branded it “a car crash waiting to happen”, as low-skilled private staff have been recruited to carry out complex work, ignoring vital local knowledge.

On Sunday, Michael Gove insisted the planned 18,000 tracers would all be recruited this week, as he put back its introduction until the end of May at the earliest.

Asked when it would be fully operational, the prime minister’s spokesperson said that this would happen “in the coming weeks”.

Asked if Matt Hancock, the health secretary, had been wrong to predict everything would be ready for mid-May, the spokesperson said: “I have only ever spoken about having the 18,000 contact tracers in place by this week.”

Insisting there was no “explicit” link between lifting restrictions and the system being in place, the spokesperson added: “We will need to look at the five tests which the government has in place and we will need to study carefully both the R rate and the rate of new infections.”