It appears that the original tune is uncertain, but Mike Raven set it some years ago. A search for Wednesbury via the onsite search engine immediately finds text and tune as posted here 5 years ago:

Lyr/Tune Add: Wednesbury Cocking

The text has since been added to the DT, but important information from the original posting has been omitted (such as the source, and fact that the text was modified by Raven) and the tune has not been "translated" into midi.

Also posted from Raven's book was a fragment from oral currency, Wedgebury Cocking, with tune. That may perhaps be useful for comparison.