A criminal investigation has been launched after a backlog of medical waste was allowed to build up at several disposal sites.
It emerged on Thursday that the waste from hospitals had not been disposed of in a timely fashion by contractor Healthcare Environmental Services.
The Environment Agency said the firm was in breach of permits.
HES said it had previously highlighted the need to invest in the UK's "ageing" incinerator network for clinical waste.
The Department of Health in England said there was no risk to the public.
A spokeswoman for the Environment Agency in England said it had found HES to be in breach of environmental permits at five sites which deal with clinical waste.
The Scottish Environment Protection Agency (SEPA) also confirmed that it issued enforcement notices at sites in Dundee and Shotts last month, where its officers are conducting "ongoing monitoring".
A Cobra meeting to discuss the issue was chaired by English Health Secretary Matt Hancock last month, according to the Health Service Journal, which first reported the story.
A spokeswoman for the Environment Agency said: "We are taking enforcement action against the operator, which includes clearance of the excess waste, and have launched a criminal investigation.
"We are supporting the government and the NHS to ensure there is no disruption to public services and for alternative plans to be put in place for hospitals affected to dispose of their waste safely."
SEPA said it was working closely with the Environment Agency and "robustly monitoring sites in Scotland".
HES has a transfer station for waste in Dundee and a storage, processing and incineration site at Shotts.
SEPA said the company had not gone over its agreed regulatory limits for either of the two Scottish sites.
A UK government spokesman said it was monitoring the situation closely and had made sure that public services – including NHS Trusts – had contingency plans in place.
He added: "There is absolutely no risk to the health of patients or the wider public.
"Our priority is to prevent disruption to the NHS and other vital public services and work is under way to ensure organisations can continue to dispose of their waste safely and efficiently."
The Scottish government said there had been no reports of any impact on services from NHS boards in Scotland.
A spokesman added: "In the event of any disruption to service at NHS sites across Scotland, NHS Scotland already has contingency plans ready to put into place to ensure there is no impact on services to patients or staff."
What is Healthcare Environmental Services?
Healthcare Environmental Services describes itself on its website as the "UK's leading independent provider of healthcare waste management".
Its latest company accounts show it had a turnover of just over £31m in the year to the end of April 2017.
They credited an 18% increase in sales to a new treatment plant in Newcastle.
Company directors Garry and Alison Pettigrew were paid £299,521 between them.
They also shared dividends of £480,000.
The company owns land and buildings worth £3.7m.
HES said it had highlighted the reduction in the UK's high-temperature incineration capacity for the last few years to the UK government, NHS bodies and the Environment Agency.
The company said all of its waste was stored "in accordance with strict regulations".
"Anatomical waste is not stored on any of our facilities for longer than is allowed by permitted guidelines," HES said.
A spokesman added: "We have welcomed visits from the EA a number of times over the past few months and have pointed out to them that the amount of waste produced by the NHS for incineration far outweighs the entire incineration capabilities of the UK, and not just HES.
"We are not the only company to feel the strain on our services, with many of our competitors continually breaching storage regulations."
Jonathan Ashworth, Labour's shadow health and social care secretary in England, called on Matt Hancock to make a statement in the House of Commons on the "staggering revelations" next week.
"When something of this scale is going on which the government has known about for some time they have a responsibility to come to Parliament and explain to Parliament whether they've got credible contingency plans and to reassure MPs who will represent those 50 hospital trusts," he told BBC Radio 4's World At One.
"Given the scale of this, it's hugely disappointing that the secretary of state has not done this," he added.
Unison's head of health, Sara Gorton, described the situation as "simply horrific", saying it was "unlikely that such a distressing situation would have happened had the service remained in-house".
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