There are calls for Britain's key workers to be given priority access to fuel, as the panic buying of gasoline across the country continues.

It is estimated that the U.K. currently has a shortage of around 100,000 heavy goods vehicle (HGV) drivers, which has disrupted deliveries and constrained the supply of goods and fuel.

In recent days, motorists have been sitting in long queues outside gas stations, often to find that no gasoline is available.

The Petrol Retailers Association (PRA) estimated on Monday that up to 90% of stations in the country were dry.

On Tuesday, Transport Secretary Grant Shapps said the situation was beginning to stabilize.

"A lot of petrol is now being transferred into people's cars and there are now the first very tentative signs of stabilization in the forecourt storage which won't be reflected in the queues as yet, but it's the first time we've seen more petrol in the petrol stations itself," he told Sky News.

However, queues and gas station closures were still rife on Tuesday.

Calls to prioritize essential workers

There are concerns that essential workers – such as those in the health and social care sectors — could soon be unable to travel to work. Dr. Chaand Nagpaul, chair of the British Medical Association, urged the U.K. government to give healthcare workers priority access to fuel.

He stressed that emergency and essential workers rely on fuel both to travel to work and for the work itself. "Everyone will have their own reasons for needing to fill up, but as pumps run dry there is a real risk that NHS [National Health Service] staff won't be able to do their jobs, and provide vital services and care to people who urgently need it," he said in a statement Monday.

Doctor-led campaign group EveryDoctor also called for steps to be taken to ensure key workers could access fuel. "It is time for the government to share responsibility for our patients' well-being by prioritizing fuel for key workers," Dr. Julia Grace Patterson, chief executive of EveryDoctor, told CNBC via email.

Meanwhile, Patrick Roach, general secretary of teachers' union NASUWT, said ongoing shortages were expected to cause "serious difficulties for the provision of education."

"For many teachers, the use of public transport is simply not an option," he said in a statement on Monday. "The Government must urgently consider making teachers a priority group for access to locally available petrol and diesel fuel supplies. Without such intervention, many teachers will struggle to get to their places of work on time."

However, not everyone is convinced that reserving fuel for certain individuals is a good idea.

Joe Armitage, lead analyst for U.K. politics at Global Counsel, said the government needed to ensure people can buy the fuel they require, "but without any draconian limits or reserving particular fuel stations for key workers."

"That creates more complexity, [and] you've often only got one member of staff in each petrol station — I don't think the practicality is viable for that," he told CNBC's "Squawk Box Europe" on Tuesday.

How has the government responded to the crisis?

Despite motorists' struggle to access gas, government ministers have insisted the U.K. has strong fuel supplies and urged consumers to purchase it as normal.

But on Monday evening, the U.K.'s Department for Business, Energy and Industrial Strategy confirmed that a limited number of military tanker drivers would be put on standby and deployed if it necessary to help stabilize the supply chain.

"While the fuel industry expects demand will return to its normal levels in the coming days, it's right that we take this sensible, precautionary step," Business Minister Kwasi Kwarteng said in a statement.

"If required, the deployment of military personnel will provide the supply chain with additional capacity as a temporary measure to help ease pressures caused by spikes in localized demand for fuel."

The government also announced on Monday that drivers with ADR licenses — which authorize them to transport dangerous goods like fuel — will be able to extend their licenses until January if they are due to expire before the end of the year.

On Sunday, Kwarteng exempted the fuel industry from U.K. competition laws, which he said would allow companies to "share information and prioritize the delivery of fuel to areas most in need." Meanwhile, thousands of truck drivers are set to be granted temporary U.K. visas in the run-up to Christmas.

Brexit's role

In a survey conducted by the U.K.'s Road Haulage Association in June, Brexit was named the biggest cause of the driver shortage by more than 600 participants, with 58% of respondents citing Britain's departure from the EU as a reason for the depleted numbers.

Meanwhile, a May report from thinktank Driver Require said the sector "had relied heavily on EU HGV drivers to avoid a supply crisis in the event of a rapid increase in demand," and claimed Brexit had "alienated EU workers."

Asked by Channel 4 News on Monday if he would send truck drivers to Britain to help alleviate the U.K.'s fuel crisis, newly-elected German Chancellor Olaf Scholz said free movement of labor was "part of the European Union."

"We worked very hard to convince the British not to leave. They decided different, and I hope they will manage the problems coming from that," he said, but added that it was important to maintain good relations between the U.K. and the EU.

Meanwhile, Michel Barnier, the EU's former Brexit negotiator, told BBC Newsnight on Monday the union's "main asset" was its single market, adding that the U.K. "has to face the consequences of Brexit."

However, Brexit is not the only contributing factor. The Covid-19 pandemic prompted many overseas workers to leave the U.K., many of whom have not returned. Changes to employment regulations have also shouldered some of the blame for exacerbating the driver shortage.

The shutdown of vocational driving tests during lockdowns, meanwhile, has prevented people from qualifying to become HGV drivers. According to the RHA, only 15,000 people were able to successfully complete training in 2020 — that's 25,000 fewer than the previous year.

The RHA also pointed out in a July report that the average age of HGV drivers in the U.K. is 55, with fewer than 1% being under the age of 25. Many drivers had retired early or found work in "less demanding sectors" during the pandemic, the organization said.

Some British employers have reportedly been offering salaries as high as £70,000 ($95,750) a year for HGV drivers, with joining bonuses of £2,000.