Each airline has their own policies regarding the time you start looking for any no-show bags. For instance each airline has their own Gate Closure before departure time and it as that point they expect you to start looking for the bags.
It could be a -15 and usually it isn't practical to do so as you haven't received clearence in some lo-cost airlines.
If you are UK based then all you need is the information on "Triple A" and that is more than what you require on process of unnacompanied luggage traveling by Air.
It is not correct that the Captain has the final say as they sometimes decide to wait for passengers. It is the Appointed Person/Dispatcher role to ensure the Safety of Passengers and an on-time departure. As soon as luggage is located and brought off the aircraft then all paperwork should be finalized and passengers should be offloaded.