Okay, now that this is possible (T263796: Watchlist query api should report time remaining), some investigation
Core entries have a clock icon between the title and the rest of the entry (including diff links) with a tooltip with the expiration
That icon can be created in JavaScript via OO.ui.IconWidget with the clock icon and shown as part of the entries

var icon = new OO.ui.IconWidget( {
	'icon': 'clock',
	'title': 'Tooltip goes here'
} );

The tooltip comes from converting the expiration timestamp into a user-facing string like "7 days left in your watchlist"
Need to implement WatchedItem::calculateExpiryInDays and ::getExpiryInDaysText in JavaScript

function getDaysRemaining ( expiration ) {
	var expirationDate = new Date( expiration );
	var differenceMs = expirationDate.getTime() - Date.now();
	var days = Math.ceil( differenceMs / 1000 / 86400 ) + 0;
	return days;
}

Use === 0 for a few hours remaining, otherwise use the same core message for days remaining.

"watchlist-expiring-days-full-text": "{{PLURAL:$1|1 day|$1 days}} left in your watchlist",
"watchlist-expiring-hours-full-text": "A few hours left in your watchlist",

Change 651706 had a related patch set uploaded (by DannyS712; owner: DannyS712):
[mediawiki/extensions/GlobalWatchlist@master] Add handling for expiring watchlist items on Special:GlobalWatchlist

https://gerrit.wikimedia.org/r/651706

Change 651706 merged by jenkins-bot:
[mediawiki/extensions/GlobalWatchlist@master] Add handling for expiring watchlist items on Special:GlobalWatchlist

https://gerrit.wikimedia.org/r/651706

doesn't work for log entries...

Change 656475 had a related patch set uploaded (by DannyS712; owner: DannyS712):
[mediawiki/extensions/GlobalWatchlist@master] watchlistUtils: pass on expiry of log entries

https://gerrit.wikimedia.org/r/656475

Change 656475 merged by jenkins-bot:
[mediawiki/extensions/GlobalWatchlist@master] watchlistUtils: pass on expiry of log entries

https://gerrit.wikimedia.org/r/656475