I don't have a lot of experience with the TH200 but in principle, is very similar to other GM units. If the transmission was working fine then and isn't now, and you changed the fluid, there are a number of things that could be causing the problem. Understanding how it works is key to troubleshooting.
The 200 has a TV (Throttle Valve) moved by the cable attached to the throttle linkage. The TV cable controls the transmission's internal "line pressure". It also has a "governor" which is a device spun by the output shaft and regulates "governor pressure" to output shaft speed. The shift valves are moved by the differential between line pressure and governor pressure. If line pressure is above governor pressure it moves the shift valve to the lower gear position. When the line pressure is below governor pressure moves the shift valve to the higher gear position.
So when taking off from a stop; you open the throttle which raises line pressure. As the car accelerates governor pressure rises. When governor pressure exceeds line pressure, the transmission upshifts. And obviously the more throttle the higher the line pressure and the more speed it takes to raise governor pressure enough to force the upshift.
Given that, if the shifts are late and hard, the TV cable is likely stuck, or miss adjusted to its wide open throttle position. This creates a situation where it will shift to 2nd at fairly high rpm regardless of throttle position and therefore you would have to turn a LOT of rpm in 2nd gear to force the shift to 3rd. So a test drive should demonstrate the 1-2 shift point. If the upshift is normal (not delayed) then its not likely the cable or a stick throttle vavle. If the problem is the cable or its adjustment, this is an easy fix. Disconnect the TV cable from the throttle linkage and check for smooth and free operation. If not, you found the problem. If so adjust it all the way in both directions and see if that effects the 1-2 shift timing.
It could be a stuck cable or TV valve in the valve body.
If its been sitting for years then it could also be that the internal seals, in this case the lip seal in the direct clutch drum, have gotten hard from age and are no longer sealing. So the TV cable/valve could be working normally and when the shift valve moves the line pressure bypasses the lip seal in the drum and doesn't apply the clutches. So it doesn't shift.
If the TV cable and valve seem to be working normally, you could try adding a bottle to Lucas transmission additive, drive it some and see if the 2-3 shift begins to work. If not, it will likely need to be gone through.