You mean laundry rooms in each flat, or communal laundry rooms? I'm sure it varies worldwide, but where I live, flat usually have a washing machine in the flat but no separate laundry room.
We use yokushitsu kansoki in most modern flats these days where I live: Laundry Lessons From Japanese Bathroom Technology - Bloomberg
The bathroom is basically a walk-in wetroom (which itself is brilliant, as they are quick to clean, space saving and accessible for those with mobility issues); they have bars hanging over the bath, and the door shuts tight with a kind of extra-tight seal. You hang laundry racks on the bars, shut the door and switch on the drying function, which then heats and dries the clothes and sucks the moisture out of the bathroom via the ventilation fan.
The UK is now quite a densely populated country and the population will not start dropping until the late 2030s, and has similar floorspace-per-person to Japan (in fact, Japan's is now slightly higher than the UK's, believe it or not), so learning from other densely populated countries like Japan and the Netherlands (including all the "little" things, like drying clothes in ways that does not cause mould issues) is the way to go.
In terms of ownership formats, England needs to learn from Scotland, where certain cities have long had "family flats" as an option for middle class families and where some excellent commonhold/freehold ownership options for flats have developed.
Balconies are a great option for families as long as they can be made safe with cages or netting if desired; you can put out laundry, paddling pools, chalk, furniture, and they are actually more private than a tiny overlooked garden. My neighbors would have lean right out and crane their necks sideways to steal a glance into my balcony. My balcony has a frosted-perspex front and laundry bars just underneath the top, so laundry can be hung invisibly, yet the balcony and flat are full of light.
I just think that it's really worth sorting out and cracking these issues, so that spacious "family flats" can become more of an option for those who would like to consider them. Because the UK's population is going to keep going up for another 15 years. If that is the case, and if everyone insists that they must have a house with a garden, the only options are "smaller and smaller rabbit hutches/subdivided houses" or "concreting over a lot more greenspace in a country that is already highly nature-depleted."