Early systems had a dedicated room in centre of the house and fans, latter systems used well insulated water tanks, but for a radiator type unit they have never been much good.
The problem is although the economy 7 and 10 gives you cheap off peak power, you also pay more for the peak power, so first thing is to decide if worth having off peak power? So first thing is define efficiency, and the normal way is to compare time taken to heat room to time the room is used. So if it takes 1 hour to heat ready to use and is used for 1 hour that is 50% efficient. Using that if a room is used for 6 hours at night and to do that it is heated for nearly 24 hours then the efficiency drops to 25%.
So if we look at other end of the scale the fan heater is very fast so likely no more than 1/2 hour before you occupy the room so efficiency goes right up. Not talking about inferred and heat pumps as then it gets really complex.
But one would think with 4 storage heaters we are looking at 4 rooms and it is unlikely you need to heat all rooms at the same time, so it is possible doing away with storage heaters could actually reduce bills and increase control, however there are other factors, I lived in a semi-detached house where the neighbour had the central heating up really high, we hardly used any gas, then the neighbour moved out and our gas bill got silly, there was it seems good insulation to outside but very poor between houses. So with flats what your neighbour does can really effect your bill and what you need to do, if neighbour has off peak power likely over night you will need no heat, and if you then go out to work, you would only need to heat 1/2 hour before you get home to approx 10 pm then the house would start to cool over night.
So in real terms only way to know is to try it. Start by same time every week reading meters and recording, borrow or buy some fan heaters, turn off the storage heaters for a week and use fan heaters instead as required, and then work out which is cheapest. Also consider is it too warm at night at the moment or too cold in the evening, once tested then you will know if off peak fits your daughters life style. There are other considerations like if there is enough power into the flat, but the basic point is, before looking at different storage radiators, work out if storage radiators suit her life style, likely young children pre-school then yes off peak good, but even when daughter at home in the day, with everyone else out I would guess she be busy and would not want house too warm anyway. And with fan heaters you only heat rooms being used. OK fan heaters not very pretty, but basic lighter the heater the quicker to heat.