When you have available "hot" instances, it just takes a few seconds.  This time also depends on how long it takes for  your application to launch and whether or not you have the "hot start" mode enabled.  When hot start is enabled, your app will already be launched and ready for a user before they connect - otherwise (with Cold Start) the app will begin loading after the user launches the session.  

If you have instances that are available but turned off (if you are using elastic scaling), it takes about 2-3 minutes during which the user will see a progress bar.