If you have night storage heating, then you can use the Economy Overnight supply to feed them. If you haven't then the tariff is probably more expensive than if you try to run e.g. dishwasher and tumble drier at night, so you may as well change to a single-rate tariff.
If you are using night storage heating, then it would be worth you paying the plumber extra for a twin-element immersion heater (they are rather expensive) and have the long element wired to cheap rate. There are dual-rate immersion timers available (I know Horstmann make some) but again they are expensive.
If the immersion heater is being paid for out of an insurance claim, then you are entitled to be provided with the same standard as you had before. if you previously had a dual immersion then you should be getting a new one. take it up with the insurance company. They might just have specified an ordinary single immersion because they didn't know. In my experience, insirance companies try very hard to be fair, and only turn mean if they think the claimant is trying to take advantage (sadly this is very common)
I think it would be best to have a separate CU for each supply you are actually using. Although dual-tariff CUs are available, they are uncommon and expensive, and people are not used to the,m and get confused on making them safe. Ask your electricity supplier what the pricing is for your tariff and make sure it is to your advantage.
p.s. sorry, my mistake, I thought you were the householder