There are usually two things that will keep that type of two-speed from shifting properly - incorrect setting of the two-speed, or a poor mixture setting. The adjustment of the two-speed is accomplished by increasing or decreasing the tension on the spring in the two-speed assembly - tightening the adjustment screw (clockwise) results in a later shift point, and loosening the screw makes the transmission shift earlier. Be sure that it's not so loose that it's shifting into second gear straight away, or that it's so tight that it can't shift at all.
Before adjusting the tranmission, however, I would be sure that you have a proper fuel mixture setting. This type of two-speed transmission requires proper acceleration as much as it does revs to enable a shift. when the engine is running poorly (too rich most often) it won't accelerate well and shift to second gear is greatly delayed, if it happens at all. So, check the main needle setting first, then adjust the transmission.