Woods In-wall Timer Heavy Duty, Model #59018 -- I just installed my timer and was able program and set the time, but when I push the manual override button to turn the light on, the light flashes and turns off immediately

Quick answer:
1) Programming steps say 'press clock key to return to current time and day'
2) Use mode button and change between AUTO & RND >> try manual override in each mode
3) Does timer turn lights on-and-off with program, or do they blink too?
4) Swap Black and Red wires
5) Try timer on light in different room, on different circuit breaker
6) Timer might be defective

Longer answer:
Woods 59018 manual does not show troubleshooting steps except 'check wiring; switch Red and Black leads; and/or reset programming'

Seems unlikely you could program timer unless timer is wired correctly ... unless there is a short of some kind.

To review wiring.
Green goes to bare ground wire.
White goes to Neutral white wire (neutral is usually 2 or 3 white wires twisted together and shoved to back of box)
Black must connect to Hot from circuit breaker
Red must connect to Load (light, fan, motor etc)

How to test wiring:
Disconnect timer and set aside.
Mark each wire
Separate wires in box so they don't touch
Use ordinary tester (you can tape tester leads to wood sticks)
Turn on power
You should have two black wires that were originally connected to switch
Test black wires to bare ground > when tester lights up, that is the Hot wire.
The other black wire goes to Load

Now you want to find the neutral.
Test the Hot black wire to white wires in box.
When tester lights up, that is neutral.

If everything checks out, I guess timer is defective.