Transmission problem
1997 BMW 540 V8 Two Wheel Drive Automatic 95000 miles

BMW 540i - Automatic -1997 mod (95000 miles). A while back started to fail to shift gears while driving. Will go into third and then not go in to 4th. It wiil only change gear at about 6000 rpm. When switching from D to Manual and switched by hand it will switch over fine. Sometimes will stay in 4th and not go to 5th unless driven up to 6000 revs or shifted manually.
Sometimes when pulling away from stand stiill it will start in 3rd and only gear back to first when full throttle is applied. When switching gears manually no problem at all. Can this be a electronical problem/programming or a mechanical problem with the gearbox itself.
The gearbox is still stock standard and has never before been opened or tampered with to change transmissin fluid and filter etc.

Without more intense diagnostics, there is only so much I can offer. If the shift body is compromised with a sticky piston or a blocked passage then perhaps a transmission service will return the tranny to full service, if not more aggressive methods can be discussed such as a remanufactured shift body or a complete rebuild of your existing shift body.