Have you been able to determine whether 12V is getting to anything else on the RAMPS board? The diode D1 possibility assumes it is. There are other possibilities if 12V isn't getting onto RAMPS. For example, a short in the 12V circuitry can cause the 5-amp polyfuse on RAMPS to open up as long as the short exists. There are two polyfuses - a 5-amp one for the electronics, motors and extruder heater, and an 11-amp one for the bed heater. The polyfuses are the large, postage-stamp like squares near the 12V inputs on RAMPS.

If you're not sure about the 12V and want to try something, one way forward would be to disconnect as much as you can from RAMPS, like the motors, the heaters, etc. I'd even consider removing the stepper motor driver boards. Leave the 12V power input connected. If the electronics powers up with things disconnected, you know the problem is in the stuff connected to RAMPS. If the electronics still won't power up with everything disconnected, and you know you have the polarity of the power supply connections right, you're down to an issue on RAMPS or the MEGA2560.