AFAIK the last machine in widespread production that handled multi-length integers was the PDP-10/20 and its clones which essentially died around 1984. I say "around" because though DEC canceled the 20 line, some clones remained (that was Cisco's original business plan, for example)
There was a series of Polish mineframes called Odra with 24bit integers which also died out in 80s essentially, but some of them were still used till 2010 in some specialized railway station software, and there was a short series of faster replacement processors for them made in late 80s-early 90s called SKOK.
Of course they weren't "widespread" for most meanings of that word :)
You can get that experience today with the C compiler for the EZ80 chip. The short is 16-bit, and the long is 32-bit, so logically the int is 24-bit. You can use it as int24_t if you prefer.
An EZ80 is used in some of the TI graphing calculators.
