The picture of the gear appears to be 56 gear.
Here is a couple pictures for comparison of a 56 and a 57 speedo gear.
The top picture is inside the speedometer hole inside the tailshaft housing of a 56 Fordomatic. You can see the little aluminum piece that the speedo gear rests upon. Have your mechanic check inside the hole with a mirror and flashlight to see if that aluminum tab has a drilled hole in it or not. The 56 tailshaft housings did not have a hole whereas the 57 housing had a hole for the aligning pin on the end of the 57 speedo gear. This will tell you if someone has changed the transmission to a later unit during its lifetime. Other ways to tell is if the trans output shaft spline is 28 or 17 spline. 28 splines are 57 and later for a Ford 292, some 56 with 312's were supposed to have 28 spline output shafts.
If everything is correct for a 56 with 292 engine and as you wrote a new cable and housing was installed along with new gear as well speedo head turns freely another thing possible is the speedo cable has not been feed properly from the transmission through firewall onto speedo head. Maybe has a sharp bend so cable is not moving freely inside the housing. Did the new cable have some light grease coating the cable for lubrication. Have your mechanic check.
marv