Ministers have been accused of losing a grip on the Covid vaccination programme for teenagers with headteachers and parents describing a “haphazard” and “incredibly slow” rollout that is causing disruption in schools in England.

They raised the alarm as Nadhim Zahawi, the education secretary, admitted he had no idea how many 12- to 15-year-olds had had their jabs, with early figures suggesting the government has little hope of hitting its target of vaccinating them all by half-term

New data has shown that fewer than one in 10 (9%) in the age group had been vaccinated by last Sunday, but this includes those who are either clinically vulnerable or living with people who are vulnerable and who were prioritised for vaccination earlier in the summer.

The government set half-term as a target for what ministers hoped would be a speedy rollout in schools, but with just over two weeks to go until the autumn break, anxious parents have told the Guardian they are seeing Covid infections rise in their children’s school but have still not been given a vaccination date.

Others said vaccinations were not being done until after half-term in November, while there are also reports of sessions being cancelled at the last minute and “poorly prepared” vaccination teams overwhelmed by demand having to leave sites after vaccinating just a fraction of pupils with consent.

According to the latest UK Health Security Agency (UKHSA) data, published on Thursday, almost 260,000 12- to 15-year-olds – out of a total of just under 3 million – had received the jab by 3 October. Of those, 94,000 have been vaccinated since the rollout began in schools on 20 September.

NHS England, however, claimed its up-to-date figures suggested more vaccinations had taken place. “In just two weeks hundreds of schools have already held vaccination clinics, with more than 160,000 children getting protected,” a spokesperson said. “As the rollout continues over the next few weeks, local providers will continue to contact schools and work with parents to agree consent so that they can organise a visit.”

Asked on Times Radio about the rates of vaccination in schools in England, Zahawi, who was previously the vaccines minister, said on Thursday he had not yet seen the vaccination data but was due to have a meeting with his former team.

He said: “It’s worth just reminding your listeners, there’s a lag because obviously you’ve got to get consent letters out, they’ve got to come back and then the school, with the school-age immunisation clinicians, decide as to when they’ve got enough students [and] parents have consented, that they can actually vaccinate.”

Labour criticised the government over the situation. Shadow education secretary Kate Green said: “Over 200,000 children were out of school due to Covid last week and the Conservatives are asleep at the wheel. Nadhim Zahawi cannot even say how many kids have had the jab.”

Dr Deepti Gurdasani, a clinical epidemiologist at Queen Mary University of London, said she was very concerned by the UKHSA figures. “Not only did the government miss the opportunity to vaccinate adolescents over the summer, even when the offer was made, it seems that most adolescents haven’t been able to access vaccination.”

Prof Christina Pagel, director of UCL’s Clinical Operational Research Unit blamed poor organisation. “In the week to 3 October, we vaccinated 70,000 children (2.4%) in England. In the three days from 1-3 October, we gave 470,000 booster doses. This is an organisational issue, not a supply issue. Scotland has already vaccinated 30% of its 12- to 15-year olds compared to our 9%.

“Cases in teenagers are extremely high. We have had months to plan a vaccine rollout to children and it is very disappointing to see it move so slowly.”

Geoff Barton, general secretary of the Association of School and College Leaders, said his members were becoming increasingly concerned about the rollout. “We have picked up reports of issues where the School Age Immunisation Service (Sais) has underestimated the challenge of vaccinating so many students at once, and has therefore had to provide vaccinations to certain year groups and arrange to return at a later date.

“We have also heard of vaccinations being cancelled because demand for the vaccine has outstripped supply, and of issues arising where either a website has crashed on which parents give consent, or the school has been asked to manage the administrative process without sufficient time to make these arrangements.”

Mark Woolhouse, professor of infectious disease epidemiology at the University of Edinburgh and a member of Spi-M (the Scientific Pandemic Influenza Group on Modelling), speaking in a personal capacity, said it was likely a substantial fraction of under-16s had already been infected with coronavirus.

“I think it’s fair to say that natural infection of under-16s is outpacing vaccination,” he said. “The good news is that the wave of infections in school-age children in Scotland is well past its peak now [according to the] ONS positivity survey data and, as you’d expect, Covid-related absences are falling too. Since Scotland’s schools went back [a few weeks] before England it’s possible that England will follow suit in the next week or two.”