"Economy 7" doesn't have to be on a separate circuit; I think that's mostly seen in Scotland and North England where electric storage heaters are more common.
Where I've lived (Midlands, South East) it's just been a cheaper rate at night, so you set the dishwasher, washing machine etc to run at 4am or whatever. It's not a cost saving unless you can do this.
That's right - all energy used during the off-peak hours is at the cheaper rate - the Economy 7 "white meter" timeswitch was just a convenience for storage heater circuits.
Where I've lived (Midlands, South East) it's just been a cheaper rate at night, so you set the dishwasher, washing machine etc to run at 4am or whatever. It's not a cost saving unless you can do this.