Scalextric start gate genie C20 pic
Can someone help me to work out why this circuit is not currently
  working?


PR should not be left open circuit (see this) and the regulator may be OK (but not if it's a 7805 because it needs a decoupling capacitor on input and output). Also your chip needs decoupling capacitors from V+ to ground if more than an inch away from any regulator decoupling.

I suppose if only one LED is used at once they can all share one resistor but be aware that trying to activate multiple LEDs will progressively dim them.

You should also get into the practise of putting a resistor in series with the base on any BJT used like this - try a 470 ohm this time but different circuits need different values.

Have you correctly programmed the device?