I was a contractor, and faced exactly this problem for a client. The client wanted an effective but inexpensive solution that didn't take a lot of interior space. My solution was the following, which the client found satisfactory:
1) I first sealed the old wall to avoid any air exchange that would transmit sound. I also went into the shopside ceiling and heavily blocked and sealed the space between joists.
2) Then I built a new studwall by turning the studs sideways and using a 2x2 for sill and plate. I firmly attached this to floor and ceiling, but floated it away from the old wall by about 1/2" without attachment. I filled this new wall space with thin unbacked insulation until it was well-packed.
3) To the new wall, I glued and screwed 2 layers of 5/8" gypsum wallboard without furring strips between.
The result was inexpensive and was very effective. No structural modifications were necessary, and the loss of floorspace in the shop was only ~3.25"
Post-script: The client had bolted his mill and lathe to the floor, but found that he had to put them on laminated wood platforms overlaying rubber pads to avoid sound transmission directly through the concrete floor to the rest of the house.