I think I like the idea of a disk directly driven by a 96 step stepper motor driven with a row of little radial sliders round the edge of it and a fixed reflective optosensor at the 12o-clock position to detect if the current slider is in or out. The U.I would be *exactly* equivalent to the mechanical timer, and all the electronics has to do is pulse the stepper once every 15 mins and drive the relay from the optosensor output, but its still eliminated nearly all the parts that wear out on the fully mechanical original.
Otherwise, what about a single button, single LED interface? push on, push off, hold 15 secs to pair with a master controller via a powerline data protocol. The master could have a full U.I. or it could be controlled by an Android device (or PC) running an app that can be used to program the timers, or fully IOT enabled. The timers would each have their own RTC so would be autonomous if they loose comms with the master.