We have a number of legacy, non-360 GTM containers that we need to upgrade to the new GA/GTM 360 version. We have done exhaustive searching on the subject and have found absolutely nothing. Hopefully someone in here has had more success with it than we have. I can't imagine that we're the only ones with this need.
After finding the right contact at Google this required upgrading the entire account to 360 via marketingplatform.google.com and editing the appropriate Account. Also requires Org Admin or Billing Admin permissions. Once this was done the containers all updated themselves.
Once it is upgraded to 360 it CANNOT BE reverted so be sure its what you want.
@ XTOTHEL above pointed to this but it wasn't at all clear. Seems that Google is not really publishing the required steps in an effort to keep it a bit more aloof / secure.