I have looked at all the pictures, but non show "the switch connected to my storage heater is currently set to turn on between 12am and 7am."
@Taylortwocities has had a guess as to what may be used, but we have not seen if this guess is correct.
Economy 7 is a tariff, not a system, it started with the "White meter" where a independent fuse board was used for off peak, we have since since then many methods, some with 5 heavy cables into the meter and the meter does the switching, some with a trigger cable from the meter, and some where the user has to do the switching.
My home uses the latter method, I have 5 hours of off peak, and there is some uncertainty as to if off peak uses BST or UTC so I only use 4 hours, my washing machine, tumble drier, and dishwasher can have a delay set in half or one hour increments, longer times one hour only, the solar inverter has a set time to recharge battery, but even if all turn on together, I would not exceed the 60 amp supply to house and the longest load is tumble drier, which takes around 2.5 hours, so 4 hours is ample.
But I will admit I had to ring up the solar panel installer to find out how to set the batteries to charge with off peak power, it was not as easy as it first looked.
However as it stands we have no idea what does the timing for you, so we can't really help.