Everyone has a spare bed set, so this gets changed on the day I wash the dirty one (aim for every week or two at most). Once cleaned and dried it returns to the cupboard in the users bedroom.
Towels get washed weekly and returned on the same day, but there is a spare towel for everyone in the house (also kept in everyone's own room, like the bedding).
Each bedroom has its own linen bin. I'm lucky my 4 year old is a tidy creature and peels his clothes off standing by the linen bin, then puts it in. The other DC is only a year old so obviously that is done for him, in his bedroom.
In my bedroom I have a linen bin with 2 compartments, 1 for whites and 1 for colours. There's a separate tiny linen bin for just work uniforms beside it.
I need to wash every day if I want to do just one wash each day. Otherwise it banks up too much on a towel or bed linen changing day, and I suddenly have to do 3 washes in a day iyswim.
Once (downstairs) a wash is done and dried, it goes into the plastic rectangle linen basket and at that point I try to pick it off the line in order of who it belongs to. That way there are 4 layers all stacked together. If in in a rush then I have to do it later.
The linen basket sits on the bottom of the stairs until the next time I am heading upstairs for any reason, then it gets left at the top of the stairs.
When I'm waiting for the bath to run for the DC I usually sort piles into each individuals clothes and put away clothes. It saves staring at water as it fills the bath and I don't seem to notice I've spent the time doing it quite so much.
I separate washes into whites, colours (both dark and light, although if it's very, very light I will allow it into the white wash), towels, uniforms and bedding. Any delicate items get out into a net protector bag and seem to manage on a mid level 40 degree wash.
The only thing I might consider doing from reading this, is trying to wash each persons clothes individually, possibly weekly, so avoid sorting clothes. Or maybe keep 4 named linen boxes downstairs so I can more easily sort it as I go once it's dried.