While I agree it seemed more like something broke, I believe it is actually more of a shift in how they handle silencing notifications. With the new Focus modes, this can be achieved fairly quickly/easily. It can also allow you more granular control over what kind of notification CAN break your session due to it's importance. Kudos to the Redditor who cued my brain to realize the new setup.
Click into the Focus settings on your phone (not sure if you can do directly on watch, but it'll be easier on phone anyway). Tap the + button to create a new Focus. There should be a pre-configured Mindfulness one there. Fill in the required information about allowing certain people/apps to interrupt or not. The last screen should automatically prompt for an option to Automatically activate when you start a Mindfulness session. Finish up, and now when you start a Breathe session for example, it should automatically switch to this focus, allowing you to not require manually switching. Once complete and you exit the app, it turned it back off for me as well. Now peaceful mindfulness sessions can continue.
FWIW - this irritated me so much at first. And of course as you often see, all online posts tend to go toward turning off the actual notifications to start a Mindfulness session - not what you desire here to stop notifications from interrupting the sessions themselves.