The fact that this happens only when first running it, and the problem doesn't happen after running it, makes me think a couple of things:
1)What kind of grease are you using on the gears? Some non hobby grease when it is cold can be really sticky, and that would fit your pattern, once the engine breaks free so to speak, the grease heats up as the gears and engine heat up, and it runs fine. I would when it was cold (hadn't been run), pull the brush plate and take out the armeture and check the grease on the end of it, if it is somewhat hard and tacky/sticky, not easily pliable, you may want to switch to a hobby grease. Also check the spiral on the end of the armature and check the gear teeth it meshes with, examine them for missing teeth or broken teeth, it could cause problems until it starts working, though that wouldn't explain why it worked when it was warm. If you can stop and start the engine every time once it is warm and it is fine, not likely to be the gearing, not to mention I would expect it would work sometime when cold...it is why I suspect the grease.
2)Potentially it could be that the commutator on the armature is dirty, and it isn't allowing the engine to create enough power to move, but once it starts moving the brushes are able to make enough connection electrically to run. I would clean the commutator and the brushes just in case that is the cause (doesn't hurt, anyway).
With stiff grease you might be able to turn the wheels easily when it is cold, but if you tried to manually turn the armature to turn the wheels it would be hard. I would recommend cleaning out the grease used in the gearbox, and replace it with new grease designed for engines, and see what happens, certainly not going to hurt and only cost is time and a bit of grease.