The no start issue is likely something else.

The relay would normally click after the timer trips to prevent the glow plugs from staying on too long.

It's easy to test. Put the car in the run/glow position on the ignition switch and then use a multimeter to test for 12v at one of the glow plugs. The 12v should go away after 30-45 seconds. If not then your timer is not working.

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