There are two reasons why the transmission won't shift into 2nd:
1) the manual shift lever is actually in 1st and not drive, although the gear position indicator shows it's in drive, or
2) the hydraulic pressure isn't building up in the valve body, which may be due to a number of factors.
The transmission in the 2001 Land Cruiser (A343F) is very similar to the transmission in the 3rd generation 80 series, and is controlled in much the same way. It doesn't have the internal control diagnostic capability that the 2nd generation Land Cruiser transmission (A750F) does. You have to find any problem yourself.
To be clear, the transmission doesn't "know" anything about itself. It's more like your oven than your PC. You give it a command and it does (or tries to) what you commanded. If it can't, the transmission control unit notes the problem and sends a message to you, in the form of a diagnostic error code.
If there is a recorded shift fault, there will be a stored error code. An OBDII reader will show you what this code is. The OBDII port is under the dash, above the accelerator pedal. You need this code to accurately diagnose and solve the problem. The recommended procedures for both finding and resolving the problem(s) are in the service manual, which is in Resources section for free download.
The 100 series service manual is much more complex than the 80 series manuals, or any manuals before them, were. This is due primarily to the fact that Toyota chose to use a computer network, known as a CANBus system, to link all the various sensors and independent computers that control the truck. There are two sections in the service manual; one is the diagnostic section and the other is the transmission section. You need to have both sections in order to properly find and fix your problem(s).
The diagnostic procedures are divided into three parts:
1) electronic problems that can be found and solved while the transmission installed,
2) mechanical problems that can be found and solved while the transmission installed, and
3) mechanical problems that require removal of the transmission to find and fix.