It emerged on Thursday that the waste from hospitals had not been disposed of in a timely fashion by contractor Healthcare Environmental Services. A criminal investigation has now been launched, the Environment Agency said.
The Scottish Environment Protection Agency is monitoring two sites where enforcement notices have been issued.
The UK government's Department of Health said there was no risk to the public.
Healthcare Environment Services, a disposal company contracted by the UK's National Health Service (NHS), is being investigated by the Environment Agency for being in breach of permits at five waste-management sites.
A Cobra meeting to discuss the issue was chaired by UK Health Secretary Matt Hancock last month, the HSJ reported.
The Environment Agency said Healthcare Environmental Services (HES) had breached its permits at five sites in England that deal with clinical waste.
"We are taking enforcement action against the operator, which includes clearance of the excess waste, and have launched a criminal investigation", the EA said.
A government spokesman said: "We are monitoring the situation closely and have made sure that public services - including NHS Trusts - have contingency plans in place".
It added it had "consistently highlighted" the issue to environmental regulators, and there has been no disruption to services to customers.
But the Environment Agency disagrees, saying that there is sufficient incineration capacity and that the majority of sites dealing with the disposal of clinical waste are in compliance with regulations.
"There is absolutely no risk to the health of patients or the wider public", a government spokesperson told CNN. "Our priority is to prevent disruption to the NHS and other vital public services and work is underway to ensure organizations can continue to dispose of their waste safely and efficiently".
The scandal was first reported by the Health Services Journal, citing leaked documents from the NHS.
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".
Healthcare Environmental Services, which is headquartered in Lanarkshire, Scotland, and West Yorkshire, England, blamed the build-up on a reduction in Britain's high-temperature incineration capacity in recent years. "This is down to the ageing infrastructure, prolonged breakdowns and the reliance on zero-waste-to-landfill policies, taking up the limited high-temperature incineration capacity in the market".
Healthcare Environment Services said the United Kingdom had experienced "reduced incineration capacity" over the previous year, which it had repeatedly highlighted to authorities.
The affected sites included a plant at Normanton - in Cooper's constituency of Normanton, Pontefract and Castleford - where the waste was reported to have been five times more than HES's permitted level.