Voltage and Current Limiting Circuit
No, the transistor circuit won't work at 20 mV, since it needs at least 0.7 V to overcome the base-emitter junction's voltage drop. It's unlikely you'll find any kind of circuit that will work at this low voltage.

But all is not lost. A current limiter may well work with a higher input voltage. It will be the load that determines the output voltage, per Ohm's Law: voltage = current x resistance. So even with a 5 V supply and a 0.2 &Omega; load you'll still have only 20 mV across it if you limit the current to 100 mA. So use a supply voltage of a couple of volts with the transistor current limiter and everything is peachy.  

Keep an eye on the transistor's dissipation. At 5 V in and 100 mA that's 500 mW and that's often the limit for a general purpose transistor. The BC337's Absolute Maximum Rating is 625 mW, for instance. So a 3 V supply is safer than a 5 V one.