NHS England says it is still offering essential vaccinations and is appealing to parents not to miss appointments for their children during the pandemic.
The childhood immunisation programme protects against diseases including whooping cough, measles and meningitis.
Visits to clinics and GP surgeries are allowed as long as none of the family is experiencing symptoms of Covid-19.
Public Health Wales said this week that it had seen a small drop in routine vaccination numbers.
Vaccinations routinely given in schools, such as the human papillomavirus (HPV) jab offered to older children, are currently suspended - but may be available from individual clinics.
"The national immunisation programme remains in place to protect the nation's health and no-one should be in any doubt of the devastating impact of diseases such as measles, meningitis and pneumonia," said Dr Mary Ramsay, head of immunisations at Public Health England.
"During this time, it is important to maintain the best possible vaccine uptake to prevent a resurgence of these infections."
Last month Unicef warned of future measles outbreaks around the world, as a result of vaccination delays due to the pandemic.
Some surgeries have taken steps to try to make the process as socially distant as possible.
The Project Surgery in East London is offering a "drive-through" service twice a week, where families can come in either by car or on foot, but do not go into the surgery itself.
It was launched when the number of routine vaccinations the surgery was doing dropped from 12 per week to just three because parents were afraid to come in.
"We chopped up the 10-minute consultation into three [parts]", GP principal Farzana Hussain told BBC reporter Anna Collinson.
"The first part is on the telephone. Then the nurse comes out just to give the injection, so the face-to-face contact is just two minutes and all the records are written up with without the patient there."
Numbers have now gone back up to eight per week, she said.
"Life is all about risks and benefits. The benefits of having your kids vaccinated is so much greater, it would be a tragedy if we saw measles or diphtheria make a comeback."
Prior to the development of the vaccine, diptheria killed about 3,500 children each year in the UK, notes Oxford University's Vaccine Knowledge Project.
It is still fatal in one in 10 cases today, but has largely been eradicated in the UK since the vaccination was introduced.