The move follows reports of at least two incidents when convoys were spotted near Penicuik, including one when the weapons were parked just yards from two schools.
Mr Thompson said he had serious concerns about the safety of driving nuclear weapons on long journeys on public roads and in highly populated areas.
He has tabled a motion at Westminster condemning the “unacceptable risk to public safety” and is to propose a Ten Minute Rule Bill this afternoon seeking to restrict the transportation of the weapons.
He said: “The idea that weapons of mass destruction are being transported through the streets of Midlothian, and all across the UK is absolutely chilling.
“One weekday lunchtime last May, a convoy was parked at Glencorse barracks for a break. It was off the road, but the barracks are next door to Beeslack High School and across the road from Mauricewood Primary School. So you can imagine pupils playing in the school grounds and just over the fence are these weapons of mass destruction.”
The convoys – tracked by campaign group Nukewatch – travel between Berkshire, where the weapons are made and serviced, and Coulport in Argyll, where they are stored and loaded.
Mr Thompson said they typically consisted of 15 or 20 vehicles in total, including extra security, and they varied their route but he claimed the risk was still “far too high”.
“The potential for an accident is huge,” he said. “The Ministry of Defence says it has an unblemished safety record, but that does not take account of the many near misses. They have breakdowns, there are pictures of them having to change tyres at the roadside and the normal things that happen to vehicles.
“When you think of what they are carrying it’s unbelievable they are on the roads.
“I’m calling on MPs of all parties to join me in condemning the unacceptable risk to public safety.
“The impact of any safety breach simply does not bear thinking about. The Ministry of Defence must now detail exactly what safety precautions they take while these nuclear weapons travel through the UK – and put an immediate stop to the convoys.”
Ten Minute Rule Bills offer MPs the chance to raise an issue of concern in the Commons chamber, but they often do not proceed much further.
Mr Thompson has also lodged an Early Day Motion, condemning the risk to public safety, calling on the UK government to clarify what safety measures they have in place and urging a stop to the convoys.
He said: “In Scotland, people have made clear our opposition to weapons of mass destruction being based on the Clyde – and the transport of nuclear weapons from one end of the country to the other should not be based on an argument of convenience at the expense of safety.
“The policy as it stands lacks transparency, is counterproductive against protecting us from terrorist attacks and shows a blatant disregard to the communities that they pass through.
“Of course, the only way to fully guarantee public safety is to remove these immoral, strategically useless weapons once and for all.”