Odds of Getting In
Your odds of getting in depend on the level of experience you have. The more FBI-like, armed security squad experience you have, the more competitive a candidate you'll be. If you've had experience in the military, you're practically a shoo-in.
Nuclear materials are very dangerous, and the people who transport them around have to be pretty hardcore. Having a good background is only the beginning. Along with your résumé, you need to fill out an Occupational Questionnaire that will determine if you are qualified for the position.
If your application is approved, the next step is the interview. If you pass that, you'll have to go through medical screening where doctors will check everything (and we mean everything) to make sure you're one hundred percent capable of doing the job. Then there's the matter of completing the Human Reliability Program.
This is a certification program where you'll go through a background check, a drug test, and a psychological assessment. You know, to make sure you don't want to use the nuclear materials for evil.
After completing your certification, the last thing between you and your shiny new job is finishing the nuclear materials courier training. We're talking about eighteen weeks of gun-shootin', daily-ten-mile-runnin', military-grade trainin'. The nuclear courier-training academy has one of the most extensive firearm programs in federal law enforcement, so it's important to get experience handling a gun ASAP (source).
At the training program, you'll also get your commercial driver's license and learn to man the big rigs. There are a lot of physical and classroom instruction milestones that you have to pass in order to complete this training. If you don't pass, you'll get the boot.