A man who returned from Italy with a fever said he waited four days for a call back from NHS 111 clinicians.
Dean Hall works on IT projects involving the British Army and told the BBC he is concerned should he have the coronavirus he might have spread it to military personnel about to deploy.
Others said the advice line gave them confusing, conflicting guidance.
The NHS said that despite high demand, all calls were being answered and the service was hiring extra call handlers.
"Anyone with concerns about coronavirus should call NHS 111 and - while the service is understandably busy and people may have to wait longer than usual - all calls are being responded to thanks to hard-working NHS staff," a spokeswoman said.
BBC health editor Hugh Pym said that some calls are coming from people who have visited places outside the list of countries at risk from the virus, adding to the pressures.
Mr Hall said he went on a five-day "whirlwind" tour of European countries with his daughter over half-term, which included Milan and nearby towns, before returning to the UK on 18 February.
He said he was sent home from work with a cough on 24 February. The next day, he called 111 when he developed a fever and the advice for travellers from Italy was updated.
'You're on the list'
Initially, Mr Hall said he was told not to worry as he returned before the cut-off date in official advice of 19 February. But on his third call after his symptoms worsened, he was promised a clinician would ring back.
By 27 February, he said he had a temperature of 102.6F (39.2C) and it took an hour-and-a-half to get through to 111. "I just got told, 'I can see you haven't had your call back, you're on the list'," he said.
Mr Hall, from Basingstoke, said he received a call from a 111 clinician on Saturday morning and he is now due to be tested for the virus.
He said his company provides IT support for the military and on 20 and 21 February he was working with the 30th Signal Regiment in Nuneaton, Warwickshire.
"If I have got it, I need to tell them and make them aware so they can go through testing before they get shipped out for deployment overseas," he said.
He said he is also concerned for his mother, who had pneumonia four times last year, while his sister and brother-in-law have both been sent home from their jobs until he gets the all-clear.
"I've literally just locked myself in my room," Mr Hall said. Trying to reduce the risk of spreading the virus to his flatmate through the shared kitchen and bathroom, he said: "I'm using more spray disinfectant than I've ever used in my life."
Carole Timms, a nurse from Shropshire who works in care homes, told the BBC she and a friend were given contradictory advice when they returned from Venice on 20 February.
Although they had no symptoms, Mrs Timms told NHS 111 she was concerned about the risk to her patients and was told to self-isolate.
But her friend, who works in a college with adults with disabilities, was told it was not necessary. That advice was changed when Mrs Timms pointed out the discrepancy.
"If she had just phoned on her own, she would happily have gone back to work when possibly she shouldn't. Or maybe I'd have got that other advice," Mrs Timms said.
The nurse said she faces losing about £2,000 of agency work while she is in self-isolation, but she said the risks with the elderly people she cares for are too great not to be cautious.
"I feel a moral duty," she said. "If God forbid, something happened, people will say, that nurse came back from northern Italy and now she's responsible for a load of deaths."
Iwona Dunsford from Milton Keynes said she contacted NHS 111 for testing after she returned from Dusseldorf with a cough and cold - although the city is not considered on the list of high-risk areas for the virus.
The advice line told her to go to Milton Keynes hospital, but the hospital referred her back to 111. Then she was sent to a chemist instead, but the chemist referred her to the hospital.
"I'm OK, but what about the people who are in a worse condition than me? On the TV we're told that the UK is well prepared for this," Ms Dunford said.
In Scotland, Alan Kelly said public health authorities initially told him to get tested after he developed flu symptoms following a trip to Milan on 14 February.
But they changed their mind when the government advice for travellers to Italy was updated, because it said only those coming back after 19 February were a concern.
"Surely the prudent thing to do would be to test people with symptoms who have been in the high risk area?" he said.
The UK chief medical officers say anyone who has returned from Iran, the 11 quarantined towns in Italy and the "special care zones" in South Korea since 19 February should call the NHS 111 helpline, stay indoors and avoid contact with others.
Anyone who has returned in the past 14 days from Hubei Province in China - where the outbreak began - should do the same.
People should also call the helpline and self-isolate if they are experiencing a cough, fever or shortness of breath - however mild - after returning to the UK from the following places:
- Vietnam, Cambodia, Laos, Myanmar and Italy north of Pisa, Rimini and Florence since 19 February
- Mainland China, Thailand, Japan, South Korea, Hong Kong, Taiwan, Singapore, Malaysia or Macau, in the past 14 days