Agree with a532942 that ventilation is key.
Mold comes from humidity in the house. The humidity is from human bodies (sweat and respiration) and from water usage (showers in particular, but any heated water that gives off steam). Humidity in the air condenses out on cold surfaces such as the interior of exterior walls, making them damp and creating the conditions for mold to grow. The colder the interior walls the bigger the condensation problem. So if the heating is used more, there is less of a condensation problem and less need for ventilation. The less heating, the colder the walls and the more condensation and the more need for ventilation. (This is often a problem in fuel-poor households: they can afford less heating and so don't want to open the windows and let what heat they have out but because of the limited heating the condensation is worse.)
The west side of the UK is more naturally humid than the east so mold is more a problem in the west. The coast is also more humid than inland, so coastal housing is more at risk than inland.
Ventilation is traditionally achieved through opening windows. With an old-style sash window, open top and bottom to get fresh air in at the bottom and hot humid water-laden air out at the top. Newer windows should have a small quarterlight at the top that can be opened for ventilation. Bedrooms should be aired out each morning to get rid of the humidity created by someone sleeping there all night, bathrooms should either have a window opened or a fan run after every bath or shower and so on, kitchens need ventilating during and after cooking and washing up.
OP is right that building quality is a big issue too. Older houses had less insulation and so colder walls, but ventilation was a lot better (windows and doors more likely to let in drafts, open fires and stoves drawing fresh air into the house), there was less indoor plumbing and less hot water use, and people lived with colder rooms so less condensation. Newer houses with good insulation have warmer walls and automatic ventilation vents in windows. The problem to avoid is an older house poorly upgraded in ways that reduce automatic ventilation but don't provide adequate insulation. Sadly, these are the houses most likely to be tenanted rather than owner-occupied, or occupied by older and poorer owners, with the consequent condensation and mold problems.