Thomas Bewick illustrated the works of Burns for William Davison and John Catnach of Alnwick.
This was a Bewick workshop production.
Iain Bain, Note on Bewick’s Engraving, p265 Memoir:
The John Thurston designs were engraved by Henry White.
“Many of the tail-pieces were from the hands of Isaac Nicholson and Edward Willis as well as their colleague White”
“But they were of the workshop and nothing would have gone out without the check and guidance of its proprietor.”
The poetical works of Robert Burns with his life
The editions are numbered TB 2.500A, 500B and 500C by Nigel Tattersfield.
Robert Burns visited Newcastle in 1787: you can read about the trip here http://community.newcastle.gov.uk/libraries/wp-content/uploads/2012/01/Robert-Burns-in-Newcastle.pdf
He did not meet Thomas Bewick however he had dinner with a "Mr. Chattox".
"Sleep at Morpeth, a pleasant enough little town, and on next day to Newcastle. Meet with a very agreeable sensible fellow, a Mr. Chattox, who shows us a great many civilities, and who dines and sups with us." [Burns Letters]
This may have been William Chatto (died 1804) the father of William Andrew Chatto, [pseud. Stephen Oliver] (1799–1864). Tea-dealer William Andrew became an important historian of wood-engraving. His Treatise on Wood Engraving, Historical and Practical (1839) included 300 illustrations by John Jackson. John Jackson (1801-48) had been an apprentice in the Bewick workshop.
You can read about Bewick's trips to Scotland and his enthusiasm for all things Scottish here http://www.bewicksociety.org/cherryburn_times/pdfs/CT%20Summer%202009%20NV-email.pdf