The original poem:
Have you any work for the tinker, brisk maids? Old brass, old pots, old kettles.
I'll mend 'em all with a tink terry tink and never hurt your metals.
First let me have but a touch of your ale, 'Twill steel me 'gainst cold weather.
Or tinkers freeze, or vintner's lees, or tobacco choose you whether.
But of your ale, your nappy ale, I would I had a firkin.
For I am old, and very very cold, and never a jerkin.
Tink could refer to the action of the tinker, or the sound made by the tinker at work. Apparently, the noise and the profession have been intertwined for some time; I found a gem of an excerpt from an 1882 book called Folk-etymology: A Dictionary of Verbal Corruptions Or Words Perverted in Form Or Meaning, by False Derivation Or Mistaken Analogy, which goes on to quote the first few lines of the 17th-century poem you cite (picture below).
Terry however, is a much tougher nut to crack. At first I wondered if it referred to the tinker polishing the kettle (as with a terry cloth) as he worked, but I don't like that theory at all. I checked the OED for terry; there were three listings, and only the first was around when the original poem was penned, and the second doesn't list terry being used as a verb:
terry, n.1 1563
...A trodden path, sometimes a baulk or ridge of earth separating fields or allotments....
terry, n.2 and adj. 1784
...The loop raised in pile-weaving (pile1) left uncut; also short for terry fabric, terry velvetB., etc., see B. In later use =...
terry, n.3 1907
...A colloquial abbreviation of territorial, applied to members of the Territorial Army...
So my other theory (which I like better), is that maybe the word is a variant of tarry, meaning the tinker is taking his time as he mends the pot, probably in hopes that the "brisk maids" break down and eventually offer him something to drink. Moreover, the OED indicates that the word tarry had several variant spellings from the era when this poem was written, including tary, tarie, and tairrie:
Still, though, that's just a theory.