The government insists the situation at petrol stations across the country is starting to improve.
But experts predict shortages could last months.
One senior minister has admitted there could be queues and closed forecourts or another 'week or so'.
Shortages are affecting private drivers and businesses.
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Home appliances firm AO has warned about delivery issues because of the shortage.
The Mirror reported that fuel industry leaders say more than one in four petrol stations are still empty.
The chairman of the Petrol Retailers Association (PRA) has warned that fuel shortages are getting worse in some parts of the country.
Brian Madderson said it remains a 'really big problem' in London and the south east.
He said he was pleased with the announcement that the military will be deployed from Monday to help out.
But he warned that will only have a limited impact.
"This isn’t going to be the major panacea," he said.
"It’s a large help, but in terms of the volume, they are not going to be able to carry that much.
"We do need a prioritisation of deliveries to filling stations – particularly the independent ones which are the neighbourhood retail sites – in London and the south east, starting immediately."
Mr Madderson said rising oil prices means prices at the pump will be higher after they're resupplied.
"Expect anything from 1, 2 or even 3p a litre increases at the pump. This is not profiteering. This is genuine wholesale price increases caused by global factors."
The key factor affecting fuel supplies is a shortage of around 100,000 HGV drivers in the UK.
This is also affecting deliveries for food and other supplies.
Experts also predict the situation may soon get worse.
There have been other warnings that meat supplies will run short before Christmas.
Ruth Gregory, of consultants Capital Economics, said: "Clouds are darkening."
Campaign group FairFuelUK said pump prices will 'definitely' rise by 5p per litre in the coming week.
Experts predict fuel prices will reach a record high within weeks.
The Company Chemists’ Association said deliveries to pharmacies were also being held up.
Rod McKenzie, head of the Road Haulage Association, said: "The driver shortage is simply not going away.
"We need urgent action."
Policing Minister Kit Malthouse told the BBC: "We are still seeing strong demand in parts of the country around fuel, albeit that there is no problem of supply into the country.
"The distribution mechanism is trying to respond to this unprecedented demand.
"My latest briefing is that the situation is stabilising, we are seeing more forecourts with a greater supply of fuel.
"Hopefully, as demand and supply come better into balance over the next few days – week or so – we will see a return to normality."
