Hi,
I would like to help you on this project of yours.
It is similar to one I had to for a medical device to treat skin for newborns.
The easier/cheap way to detect if your UV light is not working is to detect how much current is being consumed on the secondary. With a low impedance resistor you should have a small voltage drop. If there is no drop is because the bulb is burned out.
A 1USD solution is to use a reed switch. The biggest advantage is to be electrically isolated from the secondary voltage.
More expensively 2USD, you can have a optocoupler.
The timer is done with a simple RC circuit, to drive a comparator with hysteresis to avoid rapid switching.
The alarm can be either audible and/or with light. The volume, brightness (view radius) can be tweaked to met the placement of the board.
All the best,
Filipe