12-bit homebrew computer?
The baseline PIC series is very simple, has 8-bit data path and 12-bit instructions. For example, the PIC12F509. The complete instruction set can be summarized on only one page:

You can find a Verilog implementation of this core on Opencores.org

To make it simpler, you could consider 4-bits.
There were many 4-bit microcontrollers, many had 8-bit instruction widths, a few had 16, and some like the  Epson S1C60L02 had 12-bit instructions.
This particular Epson processor was produced in cooperation with a Japanese manufacturer of sensors to create the (now ubiquitous) super-cheap LCD-display thermometers based on precision NTC sensors. To top it off, they could operate from a single button cell using an internal charge pump to double the voltage. Quite impressive for 1990's technology.
If you're not stuck on 12 bits, you could also consider the 'epoch-making' (to use the Japanese-English term) TMS1000 and i4004 processors from TI and Intel respectively.