Is there a way for a team lead to see when one of their team members has completed a reminder (e.g., taken a lead out of “New” status)?

There is no summary view that shows all updates or completed tasks made by team members. However, a team leader can view all contacts with overdue reminders in their agent inbox by filtering for "contacts with reminders" and clicking on the contact to view the contact's reminder summary. 

• The Reminders filter shows up on both mobile and desktop.
• Reminders information will display in the contact panel in inbox and the contact list.