Electronic – How to generate Red light from a Blue-Dominated Spectrum

That circuit is not the best way to do this, since you will a very large capacitance for any reasonable delay. As stated in the link, 0.01F (10,000uF) is a huge amount for only 3 seconds, which will make the capacitor physically large and not so cheap. For 10 seconds you will need even more than this.

A better way is to use the RC circuit together with a MOSFET for each set of series LEDs (you can use a bipolar, but you need to take the base current required into account) in a circuit such as this one. Due to the MOSFETs almost infinite input impedance, and being able to separate the RC control from the LED drive, we can use high value resistors and therefore smaller capacitors than the RC only circuit:

MOSFET LED Delay:

Simulation:

I have used all the same LEDs here, obviously the series resistors will need to be adjusted for the desired current in your circuit. The MOSFETs can be any general purpose N-channel capable of handling the current (if it's <100mA then almost anything will be suitable). Watch the gate-sourve voltage doesn't exceed the MOSFET specs (check datasheet) - if your supply is >10V you may need to use a divider to feed the capacitor.

The turn on time is dictated by the MOSFET threshold voltage and RC time constant. You can alter this as necessary by changing the capacitor/resistor values. The timing won't be very accurate, as threshold voltages can vary quite widely, and electrolytic capacitor value tolerances are not so good, but it should be suitable for a rough delay circuit (you can tweak to suit)
If you need more accurate timing, a timer IC or small microcontroller would be better.

With the LED panels, you shouldn't need any sort of current limiting resistor or special current driver source - just connect to 12V and go. I can't say they are the highest quality, but they should work fine for what you are trying to do, although, I think the using RGB LEDs would work better as the different colors would blend better to form white than these separated panels will.

I think this could be solved very easily using many different methods. The best method for you will depend upon what you know how to do. A simple microcontroller with a couple of external buttons and transistor switches would work flawlessly, but if you don't know how to program, that is not a good route. A few logic gates driving transistors with button inputs would also work fine, but I imagine you would have already done that if you knew how.

If you would be interested in seeing a simple logic circuit, let me know and one can easily be provided.

But what about using a couple of different simple switches. Take a look at this circuit:

Switch 1 is 6 pole rotary switch. It uses rotating knob to connect the main input to one of the 6 outputs. There are many types of rotary switches with many poles (number of contacts). You will need at least a 4 pole switch, but a 6 pole switch may be better. I'll explain why in a minute.

Switch 2 is a triple-pole, double-throw switch. It has three different poles, and has two ways to "throw" them. If the switch is throw in one direction, all three LED box positive leads are connected to 12V, so they all three turn on. If it is thrown in the other direction, each LED positive lead is connected to one of the poles on the rotary switch.

If using a 4 pole rotary switch, pole 4 should be left open - this would be all LEDs off. Pole 1 would go to LED 1, pole 2 goes to LED 2, and pole 3 goes to LED 3. As you turn the knob, a different LED will be powered. There are two reasons why a 6 pole switch may be better. First, there could be an off state between each of three LED off states (1 ON, all OFF, 2 On, all OFF, 3 On, all OFF). Secondly, some rotary switches are not completely off in between states, meaning that as you turn the knob, the two adjacent LEDs will both be on for a moment, and I don't think you are wanting that to happen.

The great thing about using the 6 pole rotary switch in combination with the 3PDT switch like this is that no two LEDs can be on at the same time - it's either none, one, or all of them. The down side to using the rotary switch is that you can just select an LED bank to turn on, you might have to cycle through the colors depending on the current state of the switch.

One way to avoid this is to use a third "main power" switch to disconnect power/ground to the whole thing while you select a specific color with the rotary dial. Since you are working with color sensitive paper, this might be the best way to go.

A different way to do it is to replace the rotary switch with three common single-pole, single-throw switches like this:

With this setup, you can easily choose which LED bank to turn on. The bad thing is that you could accidentally turn on more than one at a time if you aren't careful...

Of course, no matter what you do, the switches need to be at least rated for 12V DC and as much current as the LEDs are expected to draw.

There are other mechanical switches that will select one while automatically deselecting other inputs - like how a lot of old RCA selectors used a spring loaded lever to select between your VCR, Nintendo, DVD player, etc to input into the TV RCA jacks. But again, these switch selectors typically allow more than one input to be select as they change states.