I would use "on training" as a short form of "on a training course":
- I'm on training next week.
I would use "in training" (or simply "training") for something that is more long term:
- I'm in training for the London marathon.
- I'm in training to be a nurse.
- I'm training for the London marathon.
- I'm training to be a nurse.
Would you agree?
Are there any differences between British and American English when it comes to "in training" and "on training"?