I have re-read your post, and with an instant electric shower one would not have it connected to the off peak supply, only with a stored water type would you use an off peak supply, with the latter there are two very different methods used, one is two immersion heaters the off peak is set lower in the tank to other one, and also set to higher temperature, the idea is you could leave both on and unless you use a lot of hot water the on peak immersion would not switch on.
However more common is a single immersion heater and a boost control so the timers will only normally switch on the immersion off peak, but you can if you run out of water press the boost.
However to correct for use with gas, it would have nothing to do with the meter as such, as Wilson1 says only people who can touch meter are the supply company, the tariff seem to have changed a lot, at one time there was a huge advantage using off peak with much reduced tariff at night, around 1/3 of the price, but also the on peak was more expensive so you needed a lot of off peak use to make it worth while.
However the tariffs offered have changed a lot, so today not so cut and dried, one has to shop around and in some cases the on peak is same and off peak cheaper so it is very dependent on your supplier. So why SSE would say "get an independent electrician" does not make sense. Some think has been lost in the telling of the phone call, as it does not seem to make sense.