Reloading 22 Spitfire / 5.7 Johnson rounds...

[God Bless America]

Your cases may not be fully chambering and the the bolt is not completely closed. If that is the case when you pull the trigger the hammer falls, hits the bolt, forces it closed but the firing pin doesn't strike the primer with enough force to detonate it.

Easy enough to check. Make sure the bolt rotates fully.

OP remove the bolt and check FP protrusion.

[Sailormilan2]

Which rounds did you have success with? Your home loads or the factory ammo? If you had success with the factory ammo only I would suspect a tight chamber, or a weak recoil spring. It a combination of both.
M1 Carbine recoil spring length should be 10 1/4". If it's shorter than that I would consider replacing it.
Also, once you case formed with the second step, did that involve using a sizing die? Or did you ever run your Neely created brass through a sizing die. Brass does have "spring back" and sometimes requires running twice through a die.