Electronic – One MOSFET slowly activates itself, the other won’t fully activate

You could use a quad comparator and a few resistor dividers (or a few transistors if it doesn't need to be too accurate).

You can get dedicated battery gauge ICs for this too, see for example the selection at Farnell.

Dave Jones at EEVBlog has a tutorial on using the LM3914 (for a Li-Ion, but you can see the design process and adapt as necessary)

With the quad comparator option (e.g. something like the LM339), a circuit somewhat like this should work (just thrown together, not checked thoroughly, so you will need to refine things if you choose this option - decoupling caps not shown, you will need some on the comparator rails and the zener reference):

The reference is provided by a 10V zener (D4) which is not too exact (you may want to use something more precise, notice the LED turn on values are not exact, but you can adjust the resistor values to compensate) but should do for a rough indicator. Ignore the LED part number and resistor values - they are just what LTSpice had handy (you will have to adjust the resistor values for each LEDs voltage drop to get your required current)

Simulation:

The X axis is the sweep of the supply voltage (V+) from 10V to 14V, you can see each colour LED turning on near it's respective voltage (you will need to tune for precision, as the zener reference also changes slightly with the supply voltage, or use a better reference as mentioned above)

There is a few possible scenarios. There are three parts here. The input (ie the arduino, rpi, or battery), the transistor circuit, and the output (the led channel). If you switch them around, you should be able to figure which one is the problem.

If Rpi + Green Transistor Circuit + Green led channel doesn't work, first try switching the led channel. Still don't work? Switch the transistor circuit for the blue one with the blue led channel (same rpi pin). If that works, put the blue transistor circuit with the green led channel.

If that doesn't work, then try a different RPI pin with the blue transistor circuit and the green leds.

That should eliminate any hardware issue. If it doesn't work, then it's likely software based. If the RPI pin isn't set to the right current level, or its the wrong pin, it won't work. Remember the rpi is limited to 16mA max, or less, based on setting. That said, 1k resistor means just 2.6mA at the base of the initial transistor.