I'm in training next week for your status or I'm at training next week for your location would be idiomatic to indicate that you are attending, receiving, or taking training classes, in standard English on either side of the pond. Most instances I found of he is on training and so on seem to come from non-native speakers.
But there are various contexts— albeit contrived— where on could be acceptable to reflect other ways you relate to training. There are rather numerous uses of on, after all. For some AHD senses:
1f. Used to indicate figurative or abstract position
on could be employed if training is one item in a list or sequence being done.
I am rewriting the entire employee manual. I was on travel last week, and I am on training this week.
If training is a work status or role, under
8a. Used to indicate the state or process of
you could note I was out on travel last week, and out on training this week, but I am back on duty the rest of the month.
- Used to indicate belonging to
I am on training this week could mean I am a member of the training team this week.
- At the expense of; compliments of
I am on training this week, because the QA budget ran out.
These are highly contextual or informal, and can be awkward, but they could explain the appearance of on training in various places.