as a 3rd category, next to "alerts" and "notifications".
idea is to have a 3rd button, with the number of unread watchlist items (grey if 0, some attractive color otherwise). clicking on it can either link to Special:Watchlist, or open a popup window with unread watchlist items or something.

for a "poor man's implementation" see https://he.wikipedia.org/w/index.php?title=מדיה_ויקי:סקריפטים/88.js.
the content of the popup in this implementation is incorrect and inconsistent for users who choose "Group changes by page in recent changes and watchlist"

The basic functionality of this implementation is good, IMO, except for the popup (i also used jquery.ui.dialog instead of OOjs UI) full solution needs to have a popup as pretty as the one in alerts and notifications, but the main thing is to notify about unread watchlist items.

peace.