Were there ever 12-, 24-, 48-, etc bit processors?
Yes! See https://en.wikipedia.org/wiki/Word_(computer_architecture) for an enumeration of historical word sizes.
Before the 8-bit byte became standard, computers were not byte addressable, only word addressable.
Originally, computation was mostly numeric oriented, so a (sometimes double) word data size of 24 or 36 bits was common depending on the numeric range & precision desired. 36 bits gives a decent precision in decimal digits.
Character data was stuffed into words, and handled via packing & unpacking. Prioritization of efficiency was for numerics not text.
Over time, the importance of text processing grew. The 8-bit byte became standardized, and byte addressable computers became the norm. Further, the need for interoperability of data between differing computers also required standardization. For these reasons today, it no longer makes sense to have a word size that is not a multiple of the 8-bit byte.
Your tag says microprocessors, so a quick comment on that. By the time microprocessors were developed, the size of byte was already 8-bits. As the utility of computers increased, applications became increasingly hungry for memory, so address spaces were already larger than ~32k, 64k.
Integrated circuits represented a substantial increase in performance and dramatic reduction in size, though came with the early cost of having a somewhat fixed maximum and relatively small number total transistors. These factors heavily influenced the design of microprocessors, in that they tended to have only 8-bit ALUs, though by then required 16-bit addressing and address manipulating capabilities.
Over time, microprocessors far surpassed capabilities of the old pre- integrated circuit computers (we now have 64-bit computers and RISC V has standardized a 128-bit architecture), but there was a time when things went a bit backward in word sizes before getting larger again.