The pipes in question were hot water pipes, used for district heating. I think most (probably every) house had had running cold water for decades at this point.

In contrast in the USA 7% of households were missing any plumbing in 1970.

Iceland was late industrializing and getting basic infrastructure. But so were other colonies. In 1918 Iceland stopped being a colony and by the 1950s it had pretty much caught on (with the major exceptions of road and rail networks).