After mortar is mixed it has to 'slake' in the bucket. Slake time is generally 5 to 10 minutes. If you had more than one person working I don't see why they all would be standing around - it only takes one guy to mix the mortar and the rest of the crew can be prepping the wall (removing old mortar, cleaning out the debris from the joints, and wetting the joints so the new mortar doesn't dry too fast) while one is mixing the mortar.
Once the mortar is applied, you'd clean up as much of the mortar that accidentally was spread onto the brick's surface as you could using a damp sponge. You would also shape the joints. The mortar needs to set a bit before you start on this step. One, so it can hold a shape, and two because sponging the mortar before it sets a bit can pull mortar out of the joint. Setting up a little takes time but you can use that time to start prepping the next section of wall.
After cleaning there will still be a haze around the joints. This can be cleaned up, but the mortar needs to set for a bit, otherwise over-spray from the cleaners can make it crumbly. But I wouldn't expect people to sit around waiting for mortar to dry - I'd expect them to tuck point the wall and then return the next day or after a couple of days on another job to clean up any haze around the joints. I wouldn't expect much downtime here either, as I'd work my way down the wall, and by the time I finished the place I started should be softened up enough where I could scrub off the haze.