It comes after official figures revealed 16,893 patients waited more than 30 minutes on ambulances at accident and emergency departments in England during the week up to New Year's Eve - up 42% from 11,900 in the previous week.
The Prime Minister said she recognised the situation was "difficult", "frustrating" and "disappointing".
:: Why is there always a winter crisis in the NHS?
She said: "I recognise that it is difficult for people who are facing delays. I recognise that it is difficult if someone is delayed on their admission to hospital, or if somebody has an operation postponed.
"We will hope to ensure that those operations can be reinstated as soon as possible.
"I know it's difficult, I know it's frustrating, I know it's disappointing for people, and I apologise."
It comes as tens of thousands of planned operations have been postponed as the NHS deals with the most urgent cases.
Figures published by NHS England showed the bed occupancy rate in hospitals reached 91.7% during the festive period - 85% is considered safe.
Twelve hospital trusts reported 100% bed occupancy rates on New Year's Eve.
Increased pressures on front-line services were mirrored in the NHS 111 service, which received a record 480,400 calls - up 21% from 396,300 in the previous week.
Chief executive of the Royal College of Nursing (RCN) Janet Davies said the figures showed "that almost every day last week, NHS hospitals in England were at bursting point".
She said: "Lack of beds for new patients is a major factor contributing to the current severe pressure on the NHS, but it's impossible for trusts to open extra beds without more nurses to staff them.
"The RCN has been warning of under-investment in nursing posts for several years - now that underlying problem has developed into a full-blown crisis.
"There needs to be a fundamental review of the sort of health and social care we want in this country."
:: NHS doctor: We are appalled by our own care
Mrs May spent an hour-and-a-half speaking to staff and patients at Frimley Park Hospital in Surrey, which had 99.8% occupancy on New Year's Eve.
She maintained the NHS was "better prepared for this winter than ever before" and thanked staff who worked over the season "for everything you've done".
However, Labour leader Jeremy Corbyn said Mrs May's response to the crisis was "not good enough".
He tweeted: "It is not good enough for @Theresa_May to say NHS problems are "frustrating" and "disappointing" when the Tory Government has caused this crisis.
"Nurses, doctors and patients are pleading with her to properly fund our health service, but she is ignoring them."
Shadow health secretary Jon Ashworth told Sky News the Government had been "utterly complacent' with its handling of the crisis and that "eight years of severe under-funding" for the NHS had contributed to it.
On Wednesday, Health Secretary Jeremy Hunt apologised to patients for the wave of cancellations, saying it was "absolutely not what I want".
He hit back at criticism of the Government's handling of the winter crisis from former prime minister Tony Blair, who claimed ministers' focus was on Brexit.
Mr Hunt tweeted: "Tony Blair's memory is as selective in office as out of office: does he not remember his own regular NHS winter crises? Perhaps he was too focused on joining the euro to give his full attention to the NHS..."
