The chorus wasn't in the original broadside (the text of which is quoted in another thread on the same subject; see links above). We have it only from the set recorded by Nigel and Mary Hudleston, from "Jack Pearson and men" at Filey (no date, but given elsewhere as 1957), and this is the one that was popularised by the Watersons and from which, so far as I can tell, all other Revival variants derive.

Pearson sang "down", if we are to believe the Hudleston transcription and the Waterson's understanding of the recording; and there is no obvious reason why we shouldn't. That doesn't mean that "town" is necessarily wrong (and it does make sense) but it does mean that nobody can say that "town" or even "round" is correct; those readings are interesting, but no more than guesswork and speculation.

Interestingly, the editors of Songs of the Ridings comment that "It seems that the tune is a highly corrupted 'Jingle Bells', which dates from the 1830s."

Now there's a thought. Listening to it with that in mind, I rather think I believe them.