Welcome. My name is ***** ***** would be glad to assist.
Since the timer will not activate the light in the manual mode, it does appear that one of the below is the cause:
1. The internal electronics will not trigger the relay built into the timer board.
Do you see the display show ON when you put it in manual mode or does it still show OFF
2. The relay itself has failed, and will not activate
Unfortunately, in either of these 2 cases, the timer has to be replaced, as there are no serviceable parts on the timer.
3. You can verify that there is nothing wrong with the lighting fixture, if you can remove the timer with the power OFF
then disconnect the black and blue wires on the timer from the house wiring, you will then connect the 2 house wires together sand then restore power.
If the light comes on with the power, then the fixture is fine and operable and you have proven the timer is definitely the issue.
Please mark the wires from the wall box as to which one connects to the black wire on the timer, it is the HOT feed to the timer and needs to always connect to the black wire on the timer, they cannot be reversed.
This way, you are ready for the replacement unit.
Let me know anything further needed.