4-20mA to switching output
You can make a BJT turn on at a relatively well-defined current through a base-emitter resistor. 

Let's say you choose a 12 mA (halfway between 4 and 20) threshold. We want about 700 mV at that current, which needs a resistor of 700/12 = 58 ohms, 56 is the nearest preferred value. Most general purpose BJTs will be capable of sinking the balance of 8 mA through their BE junction with no problem.

Now whether you can permit the 4-20 mA loop to be connected to your LED and power supply is another matter. If one or the other is known to be isolated, then there's no problem.

An alternative is to use the current loop to turn on an opto-isolator. Most use an IR LED at the input, with a forward voltage of about 1.3 V. A resistor of 100 ohms is then about the right value. Your opto will need sufficient current gain to light your LED directly from its output.