I once worked with an engineer whose office was located some distance from the laboratory where he normally worked, often late into the evening. His wife would call his office phone but he wasn't there to answer it, so he needed some means to notify himself in the lab when his office phone was ringing. The clever solution he came up with was a cadmium-sulfide photo-conductive sensor cell, optically coupled to one the buttons on his office desk phone (there were five illuminated button IIRC). He used the opaque red or black rubber cover from a large alligator clip (which had a standard banana jack profile on the non-gripping end) to attach the photocell to the appropriate button. The rubber cover also shielded the sensor from ambient light. He then ran a pair of wires (very small gauge!) from the photocell out through the small hole in one end of the rubber cover to his laboratory. He attached a 6V lantern battery and a doorbell, wired in series, to the pair of wires. Voila!
Office phone rings, CdS cell resistance drops, doorbell in laboratory rings.
All the phones on campus were connected via a Centrex
system, so the same button for a given line would light up on all the desk phones in a department. The phone in his lab had the bell disconnected, to avoid disturbing him from callers on the other four lines, so the doorbell arrangement allowed him to answer calls to his office phone from the lab phone.
I would modify his arrangement to include a small solenoid instead of a doorbell. The solenoid, when energized, would slightly lift the handset from between the two cradles just high enough to "answer" the phone. You could also go for a non-electronic solution using a rubber bulb (like the kind used to pump up a blood-pressure cuff) to inflate a small bladder (balloon) placed between the handset and the two cradles. I would place the inflation bulb on the floor, to be operated with your foot as soon as you hear or see the phone ringing.
so I'm going to just ask for forgiveness on this one!
Always easier to ask for forgiveness instead of asking first for permission!