It may be that MacColl sang it differently on two different records, but I don't recall hearing either. What I can tell you is that the composer E. J. Moeran noted a set from James Sutton at Winterton, Norfolk, in July 1915, which contains the following verse:

Now, I was married and my wife's name was Grey;
'Twas she that led me to shocking delay,
'Twas she that caused me to go away
On board of a man o' war, boys.

The full text and tune appeared in The Journal of the Folk Song Society, vol. VII issue 26, 1922 pp. 11-12. The text is very close indeed to that later recorded by MacColl, and he may well have got it from the Journal. Does he not indicate a source on either record?

There is some further discussion and various texts (most acknowledge no traditional source) in this earlier discussion:  Lyr Req: 'On Board Of A Man-of-war'