Education leaders have told companies not to put profit over people, claiming attempts to stop the spread of coronavirus could fail if too many parents try to keep their children in school.

Only children of key workers – including medics, police and food distribution staff – are eligible for places from Monday.

Paul Whiteman, the general secretary of the National Association of Head Teachers, urged parents to “only leave your child at school if you have no other choice”.

“My appeal to the families of key workers is: this is not business as usual. Keep your family at home if at all possible. Leave the few spaces available for those that truly have no alternative.

“My appeal to companies and other employers: please do not interpret the key workers lists liberally for your own ends. Do not put profit over people.”

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Government guidance, issued on Friday, listed the relevant occupations and children with “at least one parent or carer” who is considered critical to the Covid-19 response “can attend school if required”.

The National Education Union criticised the Department for Education’s guidance as not being clear enough and called for an urgent clarification, saying a strong message was needed to let parents know their child should only be at school if there was no alternative. Dr Mary Bousted, the NEU’s joint general secretary, said teachers were on the frontline. “They can only do this vital work if everyone plays fair,” she said.

The education secretary, Gavin Williamson, said: “Tomorrow, all schools will be closed except for vulnerable children and those of critical workers. If your work is not critical in the response to coronavirus then please keep your child at home. This will help to halt the spread of the virus, protect the NHS and save lives.

“We will be closely monitoring what is happening in schools and will ensure they get the support they need in the weeks and months ahead.”

One primary school head said it was vital numbers of children be kept low to allow social distancing. Chris Dyson, the head of Parklands primary school in Seacroft, Leeds, which has 360 pupils, has split his staff into three rotas, allowing 14-day breaks away from the classroom to allow self-isolation.

“On Friday, [after looking at the list] we were expecting 140 [pupils], but we have been texting, blitzing it on Twitter and Facebook. Because I can’t have 140. If I have 140 there will be lots of phone calls made, and by Tuesday, I won’t have more than 72, and then by Wednesday to Friday I want that number down to 36, maximum,” he said. “I hope that the message has got through. I won’t know until start of play tomorrow, but that message is out there, loud and clear.”

Jules White, the head of Tanbridge House School in Horsham, West Sussex, a secondary school with 1,600 pupils, was expecting between 90 and 100 pupils.

From his informal network of school leaders and teachers, as coordinator of the WorthLess? lobbying campaign over school funding, White said it was clear “schools are really busting a gut”.

“[There’s been] a lot of frantic work across this weekend, a lot of informal sharing … just real concern about getting it right and schools really wanting to step up, because that’s what’s happening on the ground,” he said. “I think largely parents are seeing schools in a difficult situation, and I think they are supportive.”

Simon Kidwell, the head of Hartford Manor primary school in Cheshire, and who sits on the NAHT national executive, said since the key workers list came out he had been plugging the message to parents”: “Is your job corona critical?”

He said: “First I briefed all the parents in the playground. I told them what our stance was. I used language like ‘last resort’ and ‘your family are safer’ if the social distancing is in their own home, rather than coming into our child care. We’ve had our phones on between 10am and 3pm over the weekend.”

Geoff Barton, the general secretary of the Association of School and College Leaders, said staffing would be lower because more will have to self-isolate or will become ill over the weekend. Parents would need to be prepared that not every child will get a place.

“We would appeal to parents who consider themselves to be on the key worker list only to send their child to school if there is no alternative and to treat this emergency provision as a last resort.

“We would ask them to understand that it may not necessarily be possible to provide a place for every child and that schools may have to prioritise.”