tide makes more sense, and rhymes, but as you say it's not easy to tell which is being sung on the CD- I'm pretty sure though that in singarounds 'tide' is sung.

I think the adaptation could be something to do with the 'call & response' one by one (one by one) side by side (side by side) from the 'chorus', which adds to the 'wholeness' of the song.

Or I could be talking out of my hat as I've not heard the original either :0)