That's up to the app. If you use the stock Clock app, the timer countdown and stopwatch current lap information should display on the lockscreen. It will NOT display on the Always On Display, though (it will only show the timer/stopwatch notification icon there), and you cannot display that information on the AOD, just the lockscreen.
Also make sure that you're allowing the Clock app to show content on lockscreen (App Manager, find the Clock app, tap the Notifications area and make sure the On lock screen option is set to Show content.