The very first lesson with an absolute beginner, I teach an A chord. I feel like this is the easiest chord to understand spatially and you don't have to stretch your fingers very far.
And then, we sing "Swing Low" using the one chord and doing a simple strum on the beat with our thumbs. Obviously, "Swing Low" is usually sung with at least two chords, and sometimes many more. But, it actually kind of works with the one chord.
Usually by the end of the first lesson, I can teach them an E chord as well, and we sing a two-chord version of "Swing Low."
You can also do this with "Old Blue." But in addition to that, there are some bona fide one chord songs. I have just found one that I'm planning to start using, called "Every Day," a traditional song that ukulele whiz James Hill found in a collection somewhere.