I'm in a 1 bedroom flat, all electric. I'm on an Economy 7 tariff with EDF which charges 5.34p/unit night, 11.05p/unit day. The first 238 units are 17.66p.
According to the most recent bill, this quarter I used 7.73 units/day for a total bill of about £20/month. Last quarter I used 24.93 units a day (about £50/month). The difference being not needing the storage heaters on this quarter.
If it's a modern washing machine, I suspect it will use less hot water (and therefore electricty) than washing clothes by hand. Probably less than 1 unit of electricity for a 40 degree wash.
You could halve the cost by running the washing machine at night on cheap rate electricity.
There's a table of power consumption vs energy rating here:
The immersion heater will be 3kW, so if you're running it for an hour at night that should be about 15p/day.
It sounds to me like you're not using a silly amount of electricity at the moment. It would be worth seeing if you could switch to a cheaper tariff if you're paying much more than 5p per night unit and 10p per day unit.
Regards, Richard.