Unless you have extended time away from the house, a timer on the electric WH doesn't do much. The insulation on an electric WH is normally pretty decent, and you may not be saving much of anything, and when the water cools off, can start to grow stuff you don't want.
If the timer affects anything other than the WH, it is wired wrong, and you may have a dangerous situation there that must be corrected. Impossible to say what's going on without being there to investigate.
I agree. Electrics already have fairly low ambient loss rates compared to gas (immersion coils and no heat transfer flue down the center...plus the bottom end of an electric is insulated.) The tank isn't likely to cool down much during the day. (And note that while electric resistance heating efficiency is very high, the generation of electricity is much less thermally efficient than the burner, vent, and heat transfer system of a gas water heater.)
The really critical concern with electric tank heaters is that if they are set in the recommended 120-125 F energy saving range even a few degrees of cooling (or leaving the heater off all day after a shower...then having it kick on at night) results in a temperature that is conducive to the growth of legionella bacteria. And when it warms back up to 120 F the bacteria isn't killed, it just goes dormant...takes a lot higher temp to kill it.
In the studies I've seen online of legionella, they noted that electric water heaters were much more susceptible to the problem in the samples they took. I can come up with a number of engineering reasons that this would be expected (e.g. longer recovery times, greater stratification, etc.)
The reason folks might be tempted to use a timer is if they have on peak/off peak metering. They can save money (but very little energy) by doing the heat cycles off peak. Unfortunately, that creates the potential for the unintended consequences mentioned above. Which leads me to wonder if it would be wise to run the water heater at much higher temps to compensate...a Catch 22.