If the safelight turns on when you turn the timer on then it probably isn't the fuse. If the safelight doesn't turn on then try plugging the safelight and enlarger into the AC wall plug - if they both turn on then the fuse should be replaced. If one or the other doesn't turn on then try replacing the bulbs as needed - then try the timer again. It is also possible the enlarger lamp cord has gone bad. Some enlargers have lamp switches on the lamp cord, at the base board or at the motor box - be sure the switch is turned on.
If the enlarger and safelight light up on their own, and you have put in a new fuse then the conclusion is the timer has gone bad. The timer is probably 40 years old and it's demise should be expected. There are plenty of used timers available on ebay for not much money. I would stay away from the Beseler timers - reliability isn't their strong suit. The old dial GraLabs and Time-O-Lites are indestructable. If you are going to buy a new timer then you may as well buy an f-stop model - they are cheaper than most new digital timers.