15 or 20 minutes before the screen appears?
Every time?
Have you tried both hardware resets (PRAM, and SMC resets)
Do you normally keep a monitor attached (so you have a second display), or did you just try that as part of your testing?
On that long wait, try shining a bright light/flashlight into the screen, to see if there are any icons or menus, or the dock visible (or just the login screen). Maybe it's just the backlight not working properly.
This can be the display panel, or the logic board. My bet would be on the LCD panel.