First Minister Mark Drakeford said the country has more than enough lateral flow tests.
There has been a surge in demand for Covid-19 tests as people try to comply with advice to limit the spread of the Omicron variant by ensuring they do not have coronavirus before socialising.
But by 9am on Thursday, home delivery slots for lateral flow tests were unavailable on the Gov.uk website.
Pharmacies have also complained about patchy supplies of lateral flow kits.
In a tweet on Thursday evening, he said: “Shortages of tests and delays in getting results risks exacerbating the staffing pressures in the NHS and other critical national infrastructure.
“I’ve written to the Health Secretary urging him to make sure that they’re at the front of the queue.”
New NHS England figures show 11,452 people were in hospital in England with Covid-19 as of 8am on Thursday, up 61% from a week earlier and the highest number since February 26.
The data also shows there were 2,082 Covid-19 hospital admissions in England on Tuesday. This is up 90% week-on-week and is the highest number since February 3.
The Welsh Government has agreed to loan four million more tests to the NHS in England, bringing the total the country has given England to 10 million.
First Minister Mark Drakeford said: “Wales has a significant stock of lateral flow tests, sufficient to meet our needs over the weeks ahead.”
In a letter to MPs, Health Secretary Sajid Javid said the supply of lateral flow devices (LFDs) was being tripled in January and February from a pre-Omicron plan of 100 million to 300 million per month.
“To respond to anticipated demand over the coming few weeks we are buying hundreds of millions more LFD tests, bringing new products on board and accelerating their deployment to the public,” he said.
But “in light of the huge demand for LFDs seen over the last three weeks, we expect to need to constrain the system at certain points over the next two weeks to manage supply over the course of each day, with new tranches of supply released regularly throughout each day”.
Prime Minister Boris Johnson previously urged people in England heading out for New Year’s Eve festivities on Friday to get tested.
Professor Peter Openshaw, who sits on the New and Emerging Respiratory Virus Threats Advisory Group (Nervtag), said the conditions at a New Year’s Eve gathering were “perfect” for spreading coronavirus..
“I think it’s very worrying indeed,” he told BBC Radio 4’s Today programme.
“We know the situations in which transmission happens and fortunately I don’t think we are facing the sort of lockdown that was necessary in order to cope in the very earliest part of this year.
“But we do know that crowding together in poorly ventilated spaces, particularly if you are shouting over loud music and so on, is absolutely perfect in terms of transmitting this very, very highly transmissible virus.”
In England – unlike other parts of the UK – nightclubs remain open and there are no limits on social mixing.
The NHS Covid Pass is required for entry to nightclubs but this can be obtained by people who are double-jabbed, rather than requiring proof of a negative test.
The UK Health Security Agency said on Wednesday that eight million lateral flow test kits would be made available to pharmacies by New Year’s Eve.
Professor Martin Marshall, the chairman of the Royal College of GPs, has criticised “mixed messages” over the supply of Covid tests.
Prof Marshall said the demand for tests has gone up “dramatically” as people seek to check their Covid status before socialising or coming out of isolation.
“It does seem to be that there’s some mixed messages here because the Secretary of State said yesterday that there was a global shortage because demand globally in most countries for testing has gone up massively,” Prof Marshall told Times Radio.
“But we’re also, as you say, told by the UK Health Security Agency that there’s a local logistics problem of delivering to pharmacies and delivering to the warehouses that supply the online suppliers of the testing.”
Sir John Bell, regius professor of medicine at Oxford University and a member of the Vaccine Task Force, said there was no shortage of tests but the Omicron variant had “disabled” the workforce able to deliver the tests.
He told Times Radio: “There are more than ample tests in the UK, in storage, ready to be used, and I think the problem has been the same one other people have had and that is the number of delivery drivers.
“There’s a lot of people being quarantined. The workforce which you have to have to distribute those widely has been disabled by the Omicron pandemic.”
He added: “I can promise you it is not a lack of tests.”
It comes as the NHS is setting up new Nightingale “surge hubs” at hospitals across England as it goes on a “war footing” to prepare for a potential wave of Omicron hospital admissions.
Work on a total of eight hubs, each with a capacity of around 100 patients, is set to begin as early as this week, according to NHS England.
Further sites could also be identified to add a further 4,000 “super surge” beds.
A total of 755 deaths registered in England and Wales in the week ending December 17 mentioned Covid-19 on the death certificate, according to the Office for National Statistics (ONS).
A total of 174,392 deaths have occurred in the UK where Covid-19 was mentioned on the death certificate, the ONS said.