That's a good point Roisin, the spin function really is vital.
When our last washing machine finally gave up the ghost, I washed by hand for nearly 3 weeks before the new one came, for a family of 5. The washing was no problem at all - mostly just letting things soak in the bath and agitating them now and then. But getting them 'dry' enough to hang out on the line! That was a problem. I was (almost!) yearning for the olden days of twin tubs or separate spin-driers!
I don't think many families actually 'need' a tumble-drier, most just fall into the trap of thinking they do.
We do have one, but it sits unloved and unused in our garage for the vast proportion of the year. I think we under-estimate how often we can dry stuff outside, even in the UK. I've lived in various places from the cold north-east of England to 'wet Wales' and nearly always managed to dry things outside, using an airer indoors as a back-up and to finish stuff off.
I agree with all those who have said the key is to do washing regularly. Try to catch up on your backlog, then do 1-2 loads a day, depending on your family size. I heartily agree with whoever said you need to get the message across to your family that the laundry cycle does take time so if they need that shirt/top or whatever for a certain day, you need to know that in advance.
The other thing I now do, and again I think someone else has said this, is to really check to see whether something actually needs washing. My teenage daughter changes outfit a lot and used to scoop things from the floor to the washing basket, but I've now 'trained' her to hang an outfit up again and get another wear or more out of it. Sometimes I will peg an outfit on a hanger on the washing line just to air and 'freshen' it. On a personal level, because I can be quite messy, I always wear an apron when I'm cooking or cleaning, that saves my nicer clothes from getting stained and having to be washed more. And I have some natty aprons!