I am not familiar with the Tannahill's version of the song, nor, I have to say, with Archie's (although I did hear him sing it in concert a couple of times) but what I can say is that I have known this song for over fifty years, having (literally) learnt it at my mother's knee.

First off, the second line is a mondegreen.

It isn't
..." Harris, Dunvegan, Fair Iona " ...

it's
..." Harris, or Eigg, fair Iona "

(The song is all about islands, Dunvegan isn't an island ...)

As I say , I'm not familiar with any recorded versions, but I know what my mother sang fifty-odd years ago ...

As far as the chords go, you need to realise that this is one of these songs which end on an unresolved dominant, so in fact, the chord in the first bar isn't in fact G, it's still D from the previous verse.

This leads to what I would modestly term the definitive chords for the song, which would be:

(D)Sing ye (G)o' the (Em)Cuillins o' (Bm)Skye
(F)Harris, or (Am)Eigg, (D)Fair I(G)on(D)a,
(G)Joy of my (F)heart, (C)Eilan (D)Muileach.

Whether I (G)wander (Em) east or (Bm)west,
(C)Waking or (Am)dreaming, (D)thou art (G)near (D)me
(G)Joy of my (F)heart, (C)Eilean (D)Muileach.