I agree with Richard that the problem is most likely a defective triac or solid-state relay in the timer.
Triacs and their ilk don't have a place controlling incandescent lamps unless they are protected with extensive circuitry to guard them from the currents and voltage spikes that can occur when a light bulb blows out and creates a momentary short that pulls 20 amps or more.
However, you didn't mention the enlarger head you are using with the timer, and so there is another possibility ...
If you are using a head with electronic controls or a power supply/regulator there is a possibility that a protection circuit in the timer (called a 'snubber') is letting through enough current to partially trigger an opto-isolation relay in the head. If you are using a head with a plain-ole enlarger bulb like a PH212 this won't be the case, and the symptoms the timer is exhibiting show that it is indeed broke.
If you think snubber leakage current may be the cause, try plugging an ordinary lamp into the timer and see if exhibits the same behavior. If it works fine with an ordinary lamp then you will have to disable the timer's snubber circuit.