I was told that holding my cell phone in the hand in which I have a wristwatch can cause radiation from the cell to damage the watch. It is true?"
Obviously not. This is one of those erroneous conclusions that appear in large numbers on the internet, with alarmist situations that non-technicians are unable to analyze. First, cell phone radiation is too weak to cause damage to nearby electronic circuits. Secondly, electronic watches have robust circuits and duly shielded at the points deemed necessary.
