buzz suggests something that's a really good idea, for which I've been quite criticized in the past: use old technology (capacitor charging) and don't give any suggested parts values. Still, I'd definitely try this myself.

The capacitor will probably have to be in the 4700 mfd or larger range. The resistor will make the voltage ramp up, but it will also lower the maximum light level. You might find that you have to use a 24 or higher voltage power supply because the resistor will always be in series with the LEDs. The resistor also might get hot.

A better idea is basically the same thing, but with a stage of amplification added. Charge a capacitor through a resistor, but have the capacitor voltage fed to the base of a transistor wired in an open collector configuration with the LEDs as a load. This could then be done with a much smaller capacitor.

This is more complicated because switching is not so simple since the "power" switch would turn on and off voltage to the capacitor. The actual power supply would stay on all the time.

The difference between buzz's idea and the use of a transistor is that with buzz's idea you have to make a capacitor charge up slowly to 12 volts with quite a load on it, then fade slowly, which is even more difficult without a HUGE capacitor. The transistor idea has you charging up a capacitor to only about 0.7 volts, then draining to zero. Using an FET makes this even easier as the gate current is tiny.