Is it possible to move a fully functioning internal boot HD out of one computer and install it into a new computer and have it recognize all of the new hardware? Is there anything that should/could be done to insure success?
It is possible under one condition
All the hardware in the new computer will work in that Ubuntu installation if both the computers have either the same hardware, or both computers have only the kind of hardware that does not need any proprietary drivers.
The answer depends on the hardware of the two computers. Ubuntu uses opensource drivers when possible. These drivers are part of the Ubuntu kernel. So, if both computers have all hardware that do not need any proprietary drivers, then moving the drive from one computer to the other will work.
Graphics and WiFi drivers
Graphics card manufacturers and WiFi card manufacturers don't always make public all the details of their cards, making it difficult to write opensource drivers for these cards. The manufacturers then provide proprietary drivers for their cards.
Suppose your first computer has an AMD graphics card, and the second computer has a NVIDIA graphics card. If Ubuntu installation uses AMD proprietary drivers, it won't work in the second computer. Without the graphics card being recognized the computer won't boot or will have a black screen.
Similarly, if two computers have different WiFi chips that need different proprietary drivers, then the WiFi won't work after you move the internal drive from one computer to the other.
Hope this helps