The gearbox adaption should not create any issue. They all work the same, vertical belt driven input to a horizontal chain ouput.
Earlier colts only had a fwd and reverse. Your must be a later if it was a 3 speed.
Belt engagement is seperate to the drive arrangement.
Perhaps the engine pulley has been reversed or changed.
I would not imagine that the 6 speed and the three speed would differ that much in ratio from 1st to top gear. Its the in between that the 6 speed wins out.
Bob.