Schools, nurseries and childminders were told to close their doors last week to all except vulnerable children and the children of key workers, such as NHS staff and delivery drivers.
But many nurseries say staying open for such small numbers of children has not been financially viable.
They also say staff need better protection from the coronavirus.
The National Day Nurseries Association (NDNA) estimates around half of nurseries have completely closed for the duration of the coronavirus crisis.
'Left us stuck'
But for key workers who need childcare, the closures have meant having to take time off to look after their children.
NHS consultant Vesna Pavasovic told the BBC that she would have to take time off work to look after her daughter, because their local nursery is now closed.
"We are putting our lives at risk for the entire nation and this is without any doubt our duty.
"Equally, it's the duty of the UK government to secure, without any delay, the funding for the designated schools and nurseries supporting key workers."
Catherine Phelvin and her husband, both NHS clinical workers, said the nursery of their one-year-old child closed at short notice last week.
"For various reasons and this really left us quite stuck," said Catherine.
"If we don't have adequate childcare provision for our youngest pre-school children, it's difficult for two key worker parents or indeed single parents to continue working - even their current hours, let alone, doing extra hours.
"As well as the emotional side of this, obviously it's not that easy just to switch childcare providers.
"And there's also potentially a financial cost to us as individuals, for having to find new childcare out short notice."
Eleanor Deslow also works for the NHS and her husband is a key worker in the automotive industry.
"Last Wednesday just gone, we were told that our nursery is closing - this has left us in a situation where we currently do not have any childcare."
The couple used to rely on older relatives, but government advice states children should not be left with anyone who should be following social distancing advice - for example the over-70s or those with certain health conditions.
Eleanor says her husband has now been given the option to take furlough.
"It would mean he'd be on 80% wages. But if he takes that he may not have a job to return to in three months.
"So after being promised our son would be looked after at nursery, there is no guarantee of safety being promised that the outcome will be looked after at a nursery by the government because we are key workers.
"We are now in a position, that in three months time, my husband may not have a job."
The Bright Horizons Nursery has closed "the majority" of its 300 childcare centres.
In a statement it said it had established a number of its nurseries as "hubs to help key worker families" and had waived fees.
NDNA chief executive Purnima Tanuku said a number of nurseries across the UK were opening their doors to children of key workers and vulnerable children "at this challenging time".
"However, this is causing them many difficulties which have been exacerbated by unclear guidance regarding furloughing workers."
Ms Tanuku also said staff needed greater protection from the virus: "Nursery practitioners are putting their lives on the line when they turn up to work, as it's near impossible to distance themselves from young children.
"We would like some protective equipment to be made available and acknowledgement of their dedication and commitment to these children at this difficult time."
A spokeswoman for the Department for Education said it had asked nurseries and other childcare providers to close "except for the children of critical workers and vulnerable children".
"The government has put a range of measures in place to support providers including continuing to fund free entitlements even if children are not attending, a business rate holiday for private providers, and the Coronavirus Job Retention Scheme to support workers."
If critical workers do not have access to their usual childcare place, she added, they should contact their local authority to arrange an alternative.
The government is also looking to pass emergency legislation to "require educational institutions or childcare providers to stay open".
A spokeswoman for the Welsh government added: "We are working with local authorities to identify sufficient childcare for 0-4 year olds in settings across Wales for vulnerable children and the children of critical workers."
But the Professional Association of Childcare ad Early Years raised concerns that many childminders would not qualify for the government help drawn up in response to the coronavirus epidemic.
Chief executive Liz Bayram said: "We remain especially worried about the sustainability of the 40,000 registered childminders in England and Wales.
"Most do not qualify for the current government support available to business. They are self-employed sole traders - only 3% deliver early education places so will not benefit from continuation in that funding."
Most were on low incomes and many were single parents who could only look to the benefit system for support, she added.