The pandemic has led to a sharp fall in the number of children around the world being vaccinated, the UN says.
The decline in immunisation against diphtheria, tetanus and whooping cough over the first four months of the year is the first in nearly three decades.
World Health Organization head Dr Tedros Ghebreyesus said vaccines were a hugely powerful public health tool.
He said the suffering and death caused by children missing out on vaccines could dwarf that caused by the virus.
Immunisation programmes in three-quarters of the more than 80 countries that responded to a UN survey have been disrupted, Unicef and the WHO said.
They said the disruptions were linked to a lack of personal protective equipment for health workers, travel restrictions, low health worker staffing levels and a reluctance to leave home, all of which saw programmes curbed or shut down.
By May this year at least 30 measles vaccinations campaigns had been cancelled or were at risk.
Measles outbreaks were already rising before the pandemic struck, with 10 million people infected in 2018 and 140,000 deaths, most of whom were children, according to UN data.
Unicef head Henriette Fore said the coronavirus had made routine vaccinations a "daunting challenge".
"We must prevent a further deterioration in vaccine coverage... before children's lives are threatened by other diseases, she said, adding: "We cannot trade one health crisis for another."
This will inevitably cost lives
By Richard Warry, assistant health editor, BBC News
Disruption to the global immunisation programme is extremely bad news, particularly for the world's poorest countries. It is estimated that immunisations save up to 3m lives a year by protecting children against serious diseases.
The Unicef programme is specifically targeted at children who would otherwise struggle to receive good quality health care, but although vaccines now protect more children than ever before, millions of children still go without protection, and it is estimated that more than 1.5m people die each year from diseases that vaccines could prevent.
Experts believe that low immunisation rates among poor and marginalised children seriously compromise all the gains made in other areas of maternal and child health, so major disruption on the scale outlined in this new report will inevitably cost a lot of lives.
Coronavirus has consumed huge amounts of healthcare resources worldwide as the international community has focused on efforts to combat the deadly impact of the virus.
It has also made delivery of healthcare difficult, particularly in poorer countries where supply chains have been disrupted, facilities and protective equipment can be basic, and fear of being infected has put people off attending clinics.
But diseases such as measles, diphtheria, and cholera are already on the rise, underlining the urgency of finding ways to tackle this problem.
Progress on immunisation had already been stalling before the pandemic, the UN agencies said.
In 2019 nearly 14m children - more than half of them in Africa - did not get life-saving vaccines against diseases such as measles and diphtheria.
Two-thirds of them were in 10 countries: Angola, Brazil, DR Congo, Ethiopia, India, Indonesia, Mexico, Nigeria, Pakistan, and The Philippines.
Meanwhile, historically high rates of immunisation had fallen in Latin America and the Caribbean, the UN said, with immunisation coverage falling by at least 14 percentage points in Brazil, Bolivia, Haiti and Venezuela over the past decade.
"The likelihood that a child born today will be fully vaccinated with all the globally recommended vaccines by the time she reaches the age of five is less than 20 percent," Unicef and the WHO said.
Measles outbreaks may occur as a result of the coronavirus pandemic, officials say, because some vaccination programmes are having to be delayed.
Unicef says 117 million children in 37 countries may not get immunised on time.
There have been several large outbreaks in countries across Europe where MMR vaccine uptake has been low.
The UK has already lost its measles-free status, because of rising cases of the potentially deadly infection.
The disease, which causes coughing, rashes and fever, can be prevented by two doses of the mumps, measles and rubella (MMR) vaccine, available free to all young children in the UK.
Here, 95% of five-year-olds have had the first jab - the World Health Organization (WHO) target - but only 87.4% have had the second.
And as measles is highly infectious, even small declines in uptake can have an impact.
The WHO says countries with no active outbreak of measles can temporarily pause their immunisation campaigns if necessary.
And 24 countries, including several already dealing with large measles outbreaks, have decided to delay because of the coronavirus pandemic: • Bangladesh • Brazil • Bolivia • Cambodia • Chad • Chile • Colombia • Djibouti • the Dominican Republic • the Democratic Republic of Congo • Ethiopia • Honduras • Kazakhstan • Kyrgyzstan • Lebanon • Maldives • Mexico • Nepal • Nigeria • Paraguay • Somalia • South Sudan • Ukraine • Uzbekistan
But Unicef says even more may face disruptions.
"If the difficult choice to pause vaccination is made due to the spread of coronavirus, we urge leaders to intensify efforts to track unvaccinated children so that the most vulnerable populations can be provided with measles vaccines as soon as it becomes possible to do so," it said
Spokeswoman Joanna Rea added: "Disruptions to routine vaccine services will increase the risk of children contracting deadly diseases, compound the current pressures on the national health services and risks a second pandemic of infectious diseases."
The UK continues to offer children MMR as part of its routine immunisation schedule.
Dr Mary Ramsay, Head of Immunisations at Public Health England, said: "The national immunisation programme is highly successful in preventing serious and sometimes life-threatening diseases, such as pneumonia, meningitis, whooping cough, diphtheria and measles.
"During this time, it is important to maintain the best possible vaccine uptake to prevent a resurgence of these infections."