this is a silver bullet for something that needs to be done on a specific timeframe, otherwise it'll be bad. Since the "Boss x do not give approval for it" won't cut in as a reason, and boss x needs to know this before you're doing it.

Of course criticality matters. The more critical it is, the more required for you to do a more personal message with said boss, like slack, dms, up to meeting face to face for approval.