Pretty good version and another on the Matt McGinn website: www.mattmcginn.info.
In the McGinn of the Calton book it's scored in Dm: odd couple of notes in the chorus, where they've put a Bflat, but most people, including on the 2 videos go up to the D above.
My suggestions then:

Chorus:
(Dm) We'll have a May Day my O My O
We'll have a (C)May day (Dm)then.
(Dm) We'll have a May day my O my O
We'll have a (C)May Day (Dm)then.

Verse:
(Dm) I have friends in London Town,
The finest (C)ever (Dm)seen.
(Dm) And I have some in Swansea,
And I've (F)some in Aber(A)deen.
(Dm) We're all Labour good and true,
And I'll (F)tell you what we're (Dm)going to do.
We're going to fight this fight right through
And we'll have a (C)May day (Dm)then.

I have just shown where the chord changes prior to the word where it does. If you want it lower, try Am instead of Dm, G for C, C for F, and E for A.