Those affected include candidates in Fife told they would have to wait four months and travel to test centres as far as Carlisle or Aberdeen.
The UK Government’s Driver and Vehicle Standards Agency (DVSA) said it was experiencing “unprecedented demand” for theory tests in Scotland.
Tests have been halted across most of Scotland after level four restrictions were introduced across much of the country from December 26 to January 16.
However, the agency said test centre capacity had been effectively halved because of the 2m distancing rule in Scotland compared to 1m in England.
It said Transport Scotland had also requested in October more tests for “vocational” – heavy goods vehicle or lorry – drivers to get more truckers on the road.
A Fife Labour MSP contacted by concerned constituents has alerted his MP colleagues to the problem because it is a Westminster issue.
Mid-Scotland and Fife member Alex Rowley said: "I had one person who was meant to sit her test in Stirling in the last few days and that was cancelled and she was given April in Carlisle.
"Others from Dunfermline in a similar position having to travel to Aberdeen.
"There would seem to be a large waiting time and people having to travel beyond what is reasonable.
"As you can imagine, this is a lot of costs and time to travel to these places.
"There is a test centre in Leven but it would seem it cannot cope if people are being sent so far.”
A DVSA spokesperson said: “As a result of the lockdown and the Scottish Government’s apprenticeship scheme for the logistics industry, we are facing unprecedented demand for theory tests in Scotland.
“We are offering more tests at our centres by increasing their opening hours and running tests on extra days where possible.
“We are unable to increase the number of desks for tests we provide in the current centres due to the 2m physical distancing restrictions in Scotland.
“We are sorry for the inconvenience this may cause but we ask that you are patient as we continue to protect our customers and staff from Covid-19 whilst continuing to carry out these vital tests.”
However, Skills Development Scotland said the number of its transport and logistics modern apprenticeships had followed a “broadly steady pattern” before a “significant decline” this year due to the pandemic.
Its figures showed there were 133 between April and October compared to 616 in the same period last year.
By comparison, there were 1,570 in 2017-18, 1,223 in 2018-19 and 1,452 in 2019-20.
Transport Scotland confirmed it required 2m distancing in test centres as opposed to 1m in England which “reflects Scottish Government guidance on creating and maintaining safer workplaces”.
It said in Scotland theory tests were suspended in level four areas.
The agency said it had “engaged with DVSA on securing additional capacity for vocational (heavy goods vehicle) tests following representations from vocational training organisations”.