Schedule of my storage heaters doesn't match the off-peak hours on my plan - Can you help me?

That deserves an (lol)! Anything’s possible, as we’ve discovered. 

I’ve asked around to see whether it’s possible to have a switch like the one in the last screenshot where the indicator shows whether there is power to the switch, regardless of whether it’s on or not. That would show when the switched circuit was energized by the meter’s ALCS. I did get some scary instructions involving swapping wires around inside the switch, but I chickened out of implementing them ...

I did think of the possibility that the ALCS was wrongly configured, e.g. to switch at E10 timings, but that seemed the most unlikely explanation. If the heater is wired on the constant (24-hour) circuit, then its input control could call for more charge at inappropriate times as the others have suggested. If it’s on the switched circuit, this couldn’t happen without activating the meter’s boost function to override the ALCS.

A safe way of checking the ALCS timings is provided by the little icons in the corner of the meter’s display:
  

During offpeak hours, the lower icon will change. It depends on how the switches inside the meter are wired what it changes to, but many users are seeing LC1 LC2 instead of the o‍´‍o icon. 

The change will happen a few minutes (often around 10, but theoretically up to 30) after the nominal switching times. That means staying up until half an hour after the E7 offpeak slot is due to start and looking at the meter, then looking again half an hour after the offpeak period is supposed to end in the morning. If the indicators are as expected, the meter is energizing the switched circuit at E7 times. An extra check would be to see what the icons look like when the red light comes on at the heater in the afternoon. If ıı is open as I’d expect it to be, the heater isn’t wired on the switched circuit. 

Without further evidence, I’m inclined to ​@BPLightlog’s view that the heater is switching itself on when it gets cold and the heater is on the constant circuit. If ​@VOU30  isn’t happy with this, an electrician can make the necessary changes in the consumer unit (fuse box).