Yes, time recording continues to work both at the terminals and in the app.
Depending on the terminal manufacturer, there may be different behavior if the server does not respond, but the time recording is stored locally at all existing locations until the servers respond again.
The app is designed in such a way that it saves all relevant auxiliary master data on the terminal so that it can also function offline. The recorded times are also saved locally here until a connection to the servers can be established again.