Logic OR of three wires, detect ground
A 3-input NAND or AND gate will do what you want.  For either one, there must be a separate resistor from each gate input to the +12 V.  Each gate input connects to one of the three LED pins that are pulled to GND by the controller.  The reason for the added resistors is to guarantee that an ungrounded input appears as +12 V to the gate rather than as an open circuit.
A NAND gate output will go low when all three inputs are high.  An AND gate output will go high.  This can drive a small power MOSFET that drives the white LED - anode to +12 V, cathode to the drain, source to GND.
UPDATE - Schematic
R1-C1-D1 form a missing pulse detector / lowpass filter.  The inputs to U1B go low only if all three circuit inputs (R, G, B) stay low for at least 70 milliseconds (-ish).  That is a loooong time in PWM, ans should indicate that all three LEDs are off without introducing too much delay in turning off the white LED.  You haven't given any details about the white LED, so this schematic has a generic output stage.
RESPONSE:  1. For larger white LED currents, just about any n-channel power MOSFET will work, such as an IRF520.  2. The FET can be replaced with a relay, but U1C and the FET are necessary to drive it with the correct logic polarity.  Be sure to add a suppression diode across the relay coil.  3. An MC14023 is a direct substitute for the CD part. Pin numbering should be identical; post a link to the datasheet.  4. The R1-C1 filter is necessary. Without it, the white LED will flicker constantly, varying with the brightness of the three color LEDs (the three PWM signals).