Why does the Green LED stay lit and Red LED stay off in this 555 circuit
The circuit itself is fine as long as there's enough headroom for the output to swing between 1/3 and 2/3 Vcc, and that's not a problem with a 12 volt supply and little LED loads.

According to LTspice, the circuit should work, so the only things I can think of are that maybe your wiring is different from what you've shown or you've got a bad 555.