If the spedometer is not working and the traansmision will not shift out of second it means the speed sensor is bad, unpluged or the wiring for it is damaged.
The shif6ts are controled by the ECM (computer) and if does not know the speed of the car, it will not shift.
The gear between the driveshaft and the transmission is broke you need a knew one
bad linkage in shifting cable
it usually is the gear in the transmission Could be the vehicle speed sensor.
The most common cause of a automatic transmission jerking, when shifting, is low transmission fluid. Fill the transmission to the full mark.
does the speedometer effect the transmission
gas
An automatic transmission might be slow shifting from first to second because of low transmission fluid. You might also have damaged the gears in the transmission at the area between first and second.
Is the TCM working normally and throwing no codes?
The transmission computer takes the output speed sensor signal and sends it to the cluster for the speedometer reading. The output speed sensor is a likely cause.The transmission computer takes the output speed sensor signal and sends it to the cluster for the speedometer reading. The output speed sensor is a likely cause.
Delayed shifting is because the transmission fluid is not warm enough to move through the internals of the transmission. You may need a transmission service,,,,,fluid change, new filter, etc. Transmission Fluid degrades with age.
synchonizer worn in transmission
Absence of a lubricant or so called transmission fluid.