My two cents:

A tarrier is simply one who delays. The boss arbitrarily calls the drillers "tarriers" because he thinks they're working too slow: tarrying.

If "tarrier" was a recognizable word in 1888 for something specific like a worker or anything else, it would show up somewhere outside of the song.

But as far as anybody knows, it doesn't.

The only other interpretation that is remotely plausible is that it's a dialect spelling of "terrier," a person figuratively like a terrier dog, though why this would apply to a drill operator would be another mystery. (Implying "dog" as an insult? Who knows?)

Either way, the average person in 1880s (who'd never heard of any other sort of "tarrier") might have understood the word either way and would presumably feel as uncertain abut it as we are.