I was interested in historical demographic predictions. What did the UN project for 2020 in 1950, 1960, 1970 etc.

Unfortunately pre-1950 demographic prediction is a little harder to find. I wonder why*

Anyway a reliable source is found here: https://ourworldindata.org/population-projections

In summary historical population projections have been pretty accurate but not in the way you expect.

*One thought is that because current date may become a historical inflection point similar to the first half of the 20th century, current date demographic statistics may be too important to put much stock in. Sobering.