UK poultry producers have warned that serious staff shortages caused by Brexit could mean there are not enough turkeys to go round this Christmas.

This week’s partial closure of the restaurant chain Nando’s, as well as fewer dishes on the menu at KFC, has thrown the consumer spotlight on a labour crisis exacerbated by Covid.

The British Poultry Council (BPC) said its members, which include 2 Sisters Food Group – the country’s largest supplier of supermarket chicken and KellyBronze Turkeys, had told them that one in six jobs were unfilled as a result of EU workers leaving the UK after Brexit.

The council’s chief executive, Richard Griffiths, said the group had written to the home secretary, Priti Patel, this month asking for the government to relax immigration rules but had not yet received a response.

The poultry industry employs more than 40,000 people but there are nearly 7,000 vacancies, the BPC said. The shortage means some chicken producers have reduced the size of their product ranges and cut weekly output by up to 10%, the letter said. The supply of turkey is down by a similar amount but could decline by as much as 20% at Christmas as firms fear they will not be able to draft in the usual number of seasonal workers.

Paul Kelly, the managing director of KellyBronze, which produces hand-plucked, free range turkeys, said big producers would opt to rear fewer birds if they were not confident of securing the 1,500 to 2,000 extra staff needed to pluck, pack and deliver the birds in December.

“There will be a massive shortage because companies cannot risk hatching turkeys and pushing them on the farm if they can’t get the workers to do the job,” Kelly said. “It would be financial suicide. Turkey after Christmas Day is worth nothing.”

Meanwhile, a 10th of Nando’s 450 UK restaurants are closed due to a shortage of key ingredients. The company has seconded 70 staff to its suppliers and is hopeful that all branches will reopen on Saturday.

Last week KFC said supply chain issues were disrupting its food and packaging stocks nationwide. Supermarkets have also been struggling to fill shelves amid a serious shortage of HGV drivers that is affecting areas such as milk deliveries.

“We’ve seen a loss of staff across the supply chain, particularly in our member companies,” Griffiths said in an interview with the BBC. “Our members are reporting up to 16% vacancies at the moment as a direct result of the limiting of immigration policies”

Ranjit Singh Boparan, the billionaire founder of 2 Sisters, said recently that the “pingdemic” – which forced large numbers of healthy workers to self-isolate after being “pinged” by NHS test and trace – had masked an industry already at crisis point owing to Brexit-related labour shortages.

Within its 16,000-member workforce, the majority of whom work in its chicken and ready-meal production facilities, 15% of jobs were unfilled. Brexit had “acutely reduced available workers across the food sector” with entry level jobs hardest to fill, 2 Sisters said. The company was also grappling with the soaring cost of ingredients, wage inflation and Covid-related absences, he said.

“The critical labour issue alone means we walk a tightrope every week at the moment,” said Boparan, who warned that without government help food waste would rocket “simply because it cannot be processed or delivered”.

However, the Unite union said that the crux of the problem was neither Brexit nor Covid but the “terrible pay and working conditions that make the meat processing industry one of the worst places to work in the UK”. “The conditions are awful, and the pay is worse,” said Bev Clarkson, its national officer for food, drink and agriculture.

The BPC countered that increasing wages to attract domestic workers was not the answer. “We generally operate in areas of high local employment so there is a limit to availability of local workers coupled with negligible appetite from UK workers to move from other parts of the country,” said Griffiths.