These are the faults I found:
The timer is described in the parts list as ‘a standard NE555 timer chip’. It is powered by 2 x 9V batteries. But that is the absolute maximum voltage rating for many of these chips, and exceeds the 16V absolute maximum of a Fairchild or Contek NE555.
R4 is shown shorted out so Q2 is switched hard on during the 0.2sec pulse. It cannot oscillate under those conditions.
The circuit shows on its legend ‘Page 1 of 3’, so perhaps there is more. Certainly, I would have expected to see a recommended layout to prevent possible interaction between the various inductors.
EDIT: My previous statement about problem with 555 output going high was not valid, so is now removed. I have now found details of the 555 circuitry. When output pin3 is high (sourced) and its load is connected to a voltage higher than its source voltage, the output goes O/C.
Sorry about that :-[