Open fires send a lot of hot air, their own fumes, and moisture up the chimney, and draw in cold air to replace it. The both need and provide ventilation, and that helps to some extent with damp. Condensation was still a problem.

By the late 20th century fireplaces might well have been occupied/replaced by gas fires. Unlike many modern gas fires they weren't room sealed* (i.e. you could actually touch the flames if you were stupid enough). Some vented up the old chimney, some smaller ones had no flue at all, so created moisture in the room. Air from outside was necessary whether there was a flue or not, through vents of some form, such as air bricks in the walls. My grandparents' house (Victorian) was like this is in the 1990s, with solid walls and single glazing. The windows had a lot of condensation on them in winter, but not the walls. Some of the windows in that house were louvred, and others sash; the former in particular could never be made to seal.

These fires could be fitted until surprisingly recently; I know of a house that has one in addition to gas central heating, complete with vents and chimney, and that was built about 25 years ago.

As a further intermediate step. some gas fires had a "back boiler" to heat water or even run central heating; in those cases the gas fire sat on the old hearth, with the boiler in the bottom of the chimney. I lived in a (30s-built) house in '99-02 which had hot water and rudimentary central heating done this way. That was the dampest house I've ever lived in, but also the mostly densely occupied; it struggled with condensation on the upstairs outside walls, leading to mould on things touching them long term.

* Many modern gas fires draw air through a flue in the outside wall behind them, and have a concentric exhaust. They have glass between the flames and the room, sealed to the metal housing.

Update:

There's currently a slum-housing scandal in the UK due to the inquest into death of two year old Awaab Ishak:

Awaab died in December 2020 as a result of a severe respiratory condition. In 2022, a coroner at Rochdale coroner's court ruled that this was caused by prolonged exposure to toxic black mould in his home which had "inadequate ventilation and was not equipped for normal day-to-day living activities which led to excess damp and condensation"

(A late addition, and not a historical event but I thought it would provide some context.)

Original answer:

Damp was a major problem for the UK housing stock in use during the twentieth century:

House in Thomas Street. Back to back, two up, one down (i.e. a three-storey house with one room on each storey). Cellar below. Living-room 14 ft. by 10 ft., and rooms above corresponding. Sink in living-room. Top floor has no door but gives on open stairs. Walls in living-room slightly damp, walls in top rooms coming to pieces and oozing damp on all sides.

George Orwell - The Road to Wigan Pier (first published 1937)

There was a huge expansion in house building during the nineteenth century driven by the need to house workers during the industrial revolution. Construction was often cheap, quick and to poor standards which resulted in damp, mould, insect infestation (cf. The Road to Wigan Pier) and poor sanitation. Many of these buildings continued to be used, unimproved, well into the twentieth century. The lack of regulation was first addressed by the Public Health Acts of 1875 and 1878, and 1878 Building Act.

From the 1890s on, large scale slum clearance has been undertaken in an attempt to replace poor housing with new development, however improvements in accommodation often came at a cost of damage to communities.

It would be a mistake to assume that damp and mould are historical problems that are no-longer found in UK housing:

Tenants of one of England’s largest housing groups say they have been left for years in ‘uninhabitable’ homes as experts warn social renters could be left with long-term health conditions as a result of damp and mouldy properties.

From INews 28 March 2022

And also:

As a nation, Britain has a big problem with damp, it can not only cause damage to your home but if left for a long period of time, it can also cause problems to your health as well. According to Shelter, there are 1 in 5 renters who are affected by damp and mould in Britain. So why are mould and damp problems so common in British homes?

Timberwise, 25 January 2022