I've played a game today where I built a farm on top of a river that was already producing 1/1/1. After the farm was finished, the yield was unchanged? What gives?

The farm should have added 1 nutrition. As it was probably a moist square (they usually are when there's a river) the base output should stack with a farm. The river itself only yields +1 energy. Did you build anything else there, that would reduce nutrition output, e.g. a mine?

Nope, nothing but farm. I tested it again quickly on a new map. Same thing. It was Gaians both times if it matters.
And both times with kyrub's patch, if that matters.

Rivers don't do anything other than give +1 energy, act as a road and sometimes create 'sink holes' during global warming.