In an unprecedented move on Tuesday, NHS medical director Sir Bruce Keogh ordered the cancellation of all non-urgent surgery until at least February.
"As with the majority of other hospitals nationwide, ELHT's Emergency Care Department has been experiencing a surge in demand over the Christmas and New Year period".
Dr Nick Scriven, the president of the Society for Acute Medicine (SAM) and a consultant at a hospital in the north of England, worked on Monday from 8am until 7pm.
Thousands of operations and procedures in England's National Health Service may be postponed until January 31 because of the influx of emergency cases.
It is too early to judge how bad this year's pressures are compared with previous years.
One patient took to social media to air his grievances, claiming he spent an "undignified" 18 hours on a trolley waiting for a hospital bed.
"We will prioritise the sickest patients" over non-emergency visitors to A&E, health bosses have said as winter pressures set in.
"I am extremely proud of the outstanding service they provide and can not thank colleagues enough for their dedication, additional hours worked and the care they provide to each other, as well as our patients, when the going gets tough".
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The board's office-based employees were asked to consider volunteering to help their colleagues in hospitals and GP practices by taking on cleaning and administrative roles over the next five days. "This is the busiest week of the year for the NHS".
NHS England reiterated that cancer operations and time-critical procedures should go ahead as planned.
The MP for South West Surrey also paid tribute to the NHS staff working in "very tough conditions".
Ms May said the NHS was "better prepared for winter than ever".
"We also thank members of the public for their continued support and continue to request they only attend A&E if absolutely necessary".
"We recognise that there will be upset and inconvenience caused to patients and their families, and we really appreciate their patience and understanding during this hard time".
There would also be more resources for ambulatory care and clinics as alternatives to hospital admissions, and a twice-daily review of all patients to facilitate discharge.
Attending booked appointments is a "small but effective" way to help, she said.