Problem with delay before LED turns on
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 &lt;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.