Computers don't need to heat up. They do that naturally which gets controlled by fans or heatsinks.
Are you saying it takes 15 minutes for the display to turn on? Do you see the POST (Counts your RAM, detects HDDs etc.) information when you power up the laptop? Can you see the Operating System loading screen? Where does it fail exactly?
Do the fans spin and any LED indicator lights light up immediately when you push the power button to boot it? If it's just your screen that is not displaying right away then that could be an inverter problem, but if the system is taking 15± minutes to make it through POST, that's another story entirely.