Think you're right about those chords. I found another video of Brett playing the same song on guitar.
Although he uses the capo at the 4th fret (rather than the 5th that would allow the last four strings to match a uke), you can see that his chord shapes for the guitar is C, G, F (possibly Fmaj7), and I think there's a Am shape as well. So the uke version should be F, C, Bb (or Bbmaj7), and Dm.