Currently 'Boost for Lemmy' doesn't show any real-time notification. The minimum check interval can be set to 30 minutes and that's also inconsistent. There should be an option or by default it should show real-time notification. This issue was also present in 'Boost for Reddit'.
The app has to make a request from your Lemmy instance to get notifications. For "real time" notifications it would have to be requesting updates constantly, which wouldn't be appreciated by instance owners and could maybe even get you blocked.