Hi esetnik,
Thanks for posting to Apple Support Communities. It sounds like you have to switch the Bose headphones off and back on in order to hear out of them instead of the iPhone.
Make sure your iPhone is updated to iOS 12.1.4 which was just released today: Update your iPhone, iPad, or iPod touch.
Next, run through the troubleshooting listed here to see if these steps alleviate the issue: If you can't connect a Bluetooth accessory to your iPhone, iPad, or iPod touch
Are you able to see how your Bose Headphones work with another iPhone? If the same issue occurs, this may be a problem with the headphones.
Check to see if a firmware update is available: Bose® SoundLink® around-ear Bluetooth® headphones update (Formerly: AE2w Bluetooth® headphones)
Also, if you check here: SoundLink around-ear wireless headphones II there is an app available for the headphones inside the App Store that could help. "The Bose Connect app gives you easy access to everything your product has to offer. Get tips, unlock product features and personalize your settings, so you can spend more time enjoying the music instead of setting it up."
Have a fantastic day!