Economy 7 has a single dual-rate meter connected to a single set of electrical wiring in the property. (I had this in the 1980s.) To benefit from the cheaper night rate your storage heaters will need a timer somewhere so they are only switched on during the night rate period.
Back in the 1960s I vaguely remember my (English) grandparents had a different system then known as a white meter which had a second (white!) meter for night rate heating. This was in addition to the standard day rate meter which was in the usual black Bakelite case.
The white meter came with a contactor feeding a separate set of electrical wiring which was only connected to devices on the night rate. Storage radiators would be connected to this wiring. They didn't need their own timers as the white meter/contactor would only switch them on during the night rate period.
There is a diagram of the two systems at this link (scroll down):
Note this original white meter system with separate wiring has disappeared in favour of Economy 7, which is cheaper to install. In Scotland Economy 7 is known as White Meter and this usage has now spread south of the border. As the change- over to E7 occurred pre-WWW it's hard to find links to the original white meter system. A lot of modern links to "white meter" are just synonyms for E7.