The warning comes after days of long queues at petrol stations amid panic buying triggered by supply problems that left some service stations running dry.
GPs have raised concerns that without fuel, healthcare staff and other key workers are struggling to get to work - while some patients are already reported to have been unable to attend appointments because of a lack of petrol.
BMA chair Dr Chaand Nagpaul said: 'Emergency and essential workers rely on fuel both to travel to work and for their work itself – whether this is to get to hospitals, practices and other healthcare settings, or for ambulances to reach people in urgent need of care and GPs to visit very ill patients at home.
'Everyone will have their own reasons for needing to fill up, but as pumps run dry there is a real risk that NHS staff won’t be able to do their jobs, and provide vital services and care to people who urgently need it.
'While the government has said it is putting plans in place to alleviate the shortage of HGV drivers to transport fuel, the results of this won’t be immediate. Healthcare and essential workers must therefore be given priority access to fuel so they can continue their crucial work and guarantee care to patients.'
In a statement on 24 September, energy company BP warned that it was 'experiencing some fuel supply issues at some of our retail sites in the UK and unfortunately have therefore seen some sites temporarily close due to a lack of both unleaded and diesel grades'.
In an echo of problems that have affected flu vaccine deliveries and supplies of blood tubes to NHS in recent weeks, the company blamed the problem on 'delays in the supply chain, which has been impacted by the industry-wide driver shortages across the UK'.
The government has announced emergency measures to tackle the shortage - freezing competition rules to allow providers to collaborate to reduce disruption to fuel supplies.
Minsters have also announced measures such as an immediate increase in HGV testing, short-term visas for HGV drivers and 'bootcamps' to train up to 3,000 more people to become HGV drivers.
Business secretary Kwasi Kwarteng said: 'We have long-standing contingency plans in place to work with industry so that fuel supplies can be maintained and deliveries can still be made in the event of a serious disruption.
'While there has always been and continues to be plenty of fuel at refineries and terminals, we are aware that there have been some issues with supply chains.'