1990 chevy Lumina has speedometer, not recording Mileage and delay from shifting 1-2 and 2-3. I think it is the transmission speed sensors. Any other suggestions?

Yes

There is something not connected, What was the original issue?

could be the VSS - vehicle speed sensor .
Speedometer and Odometer
The Vehicle Speed Sensor (VSS) is a gear driven permanent magnet generator. The vehicle transaxle houses the VSS. The VSS generates a sine wave output with a frequency proportional to the vehicle speed.
The Powertrain Control Module (PCM) converts the CKT 389 signal to an output that switches to ground at a frequency of 4000 pulses per mile. The CKT 389 signal drives the speedometer and the odometer. Various other vehicle components use the CKT 389 signal. Refer to Engine Controls.

Go to online store reliable they have repair manuals I am sure you will get the solution. Here's the link
Best Manuals

YEP YOU NEED VSS SENSOR HOOKED UP SO TRANSMISSION CAN SHIFT RIGHT.AND SHIFT TO OVER DRIVE AT SET SPEED. YES YOU NEED TO SEE YOUR SPEED TO KEEP FROM GETTING A SPEED TICKET.DONT UNPLUG IT.

If it won't go into overdrive a speed sensor within the transmission may not be sending the shift signal to the transmission. Check fluid levels to make sure that's not causing the problem. Also, check that the fluid is red color and clean.

Transmission probably needs to be serviced at a reputable shop if you want this problem fixed.

have the shift solenoid checked out
Auto Zone will perform a free scan of the computer to dee which one is at fault

Either the gauge is toast...or more than likely, your cable is having problems at the tranny. Check for damage to the cable at the transmission housing. And, if you have speed ratio attachment where the cable goes into the tranny....that might be going bad.

Check the signal from the Vehicle Speed Sensor with a scan tool, that is the most likely cause.

Your car has the 125c/3T40 model trans,3 speed automatic with lockup.When you say you unplugged the "speedometer",if you are referring to a square electrical connector on the trans,facing the radiator,that connector is not the speedometer,but the lockup circuit inside the transmission.It is likely the lockup solenoid inside the side pan has gone bad and is malfunctioning.It is a common problem on those transmissions.You will lose some gas mileage by unhooking the connector,and eventually,years down the road,will hurt the trans from the fluid running too hot.Lockup clutch application in a General Motors lockup torque convertor helps to cool the fluid.