I did a Google for storage heater wiring, and it seems there are many versions, wired in different ways. Some do have both peak and off-peak supplies, but the peak supply may be just for a fan, or it could have special boost elements, some have built in time clocks, others don't, so unlike the days of the original "White" meter, one needs to look at each storage heating system independent to rest, as there are so many variations.
Most will have two meters

This is the problem, there is no one fits all, here clearly one meter, and not all old set-ups used

RTS, there are still a few around it seems with simple timers.
The same problem with Smart meters, we have a whole array of systems, some the Smart meter has two outputs peak and off-peak, some it is down to the user to ensure his equipment turns on at the right time.
Here was are responding to
@DiyNutJob we are not really worried about any other systems, only his. He asks "Does the RTS turn off at the end of June affect lights and sockets?" simple answer not it does not.
@Mottie made non-useful comments, but
@Taylortwocities in #3 gave the answer. Plus extra advice and a request for pictures to confirm.
I rambled on, maybe I should not have? And
@DiyNutJob posted pictures of his meter.
@Taylortwocities again very good post referring to the bill, and asked for more pictures.
@Rodders53 came in with some good background information, and
@Harry Bloomfield repeated
@Taylortwocities for more pictures and also like
@Taylortwocities referred to the bill.
@endecotp pointed out one could be over paying if times are wrong, and
@aveatry referred to repeated adverts and letters, and
@fixitflav and
@Harry Bloomfield talked about rumours as to what may happen.
@flameport then explained, as only he can, what is inside the units, and pointed out it is a very real problem, and I pointed out there were threads where this had been talked about in more general terms.
But we have not seen a picture of the consumer unit/s, however it does seem as far as
@DiyNutJob is concerned unlikely to lose power, the only question is will he be overcharged. No real answer can be given until we get more information, it will be interesting to see what the suppliers do if someone is overcharged due to them not changing the meter on time, but I would think nothing, when I look at my bills.