[Blue Bowtie]

If the transmission is holding second without going to third, it could "live" indefinitely. That would indicate that the valving is holding second and not applying third. If the transmission is going from second to "no-man's land" and then finally engaging solidly third, that could indicate a valving or actuator problem not engaging a clutch, or not releasing the band, in which case the transmission may have a short time left. About all you could do externally (short of a scan) is measure resistance of all the solenoid coils (from the outside) but this will only tell you that they are operational. Apparently, they are, since it evidently finally does shift. You might also check line pressure with an external gauge to verify that the pump is operational and line pessure is correct in the various ranges.

You really need to monitor the transmission pressure sensor/switch data while the trans is in operation.

Another thing you can do as a general test is to operate the transmission manually through the ranges to see if it shifts when the selector is moved, or delays.

Also, does it engage both REVERSE and DRIVE normally, without delay?

A Snap-On Solus MT-2500 can read all powertrain parameters with the correct personality module and keys.