Where does Shakespeare write "woollen pipes"? I'm not throwing doubt on the quote - but it'd be interesting seeing the context.

Given that uilleann pipes just means elbow pipes I'd be a bit surprised if the term hadn't been used for pipes with bellows - I mean, what else would you call pipes worked by the elbow (assuming you were speaking Irish)?

And then when the "union" pipes came along, using the term for this splendid innovation would have been an obvious thing to do.