If you look on General Config, you will see that you can alter what action Mach3 takes on a tool change.
The default is "Ignore Tool Change". If you type in the M6 followed by T*** and alter the config - in your case I think I would use the "auto tool changer" option, becasue the other alternative stops the machine to let you change tools.
If, on your tool table, you put in accurate offsets for the three spindles, along with the tool lengths, the machine should then move to the new spindle.