Hi,
The circuit won't work with one transistor because for a circuit to work as an oscillator there should be a positive feedback present which is only possible using two transistors which cause alternate charging and discharging of the two capacitors and thereby turning the transistors alternately.
If you want only one LED to blink, you can use the same circuit but just replace the other LED by a small resistor of 470 Ohm or so. Thus the circuit will have only one LED.