I am interpreting "processing" as action in the Develop Module. Drive speed doesn't make more than a trivial difference in the Develop Module. So put your photos on a regular spinning disk, even an external spinning disk. The drive that Lightroom is stored on also doesn't make the slightest bit of difference to Develop Module speed (although LIghtroom will load faster if it is on a fast disk). CPU speed and GPU (if enabled) speed are the primary factors that influence Develop Module speed, but other things such as image size, monitor size and the amount of local adjustments also affect Develop Module speed.
All things being equal, I would put the catalog and its caches on the fastest internal drive you have that is not the system drive.
That said, all things are not equal. Between the OS and the SQLite database and LR, there is enough disk caching going on that drive technology does not give any human-noticeable performance differences.
I find that for drive performance, things seem to work happiest with the OS and LR catalog and image files all on separate drives. That way you are usually not trying to read and write to the same drive at the same time during large I/O operations.
For processing speeeds, CPU and memory are the things to be looking at. Sometimes the GPU can help you, sometimes not.
That way you are usually not trying to read and write to the same drive at the same time during large I/O operations.
I'm not sure how this applies to Lightroom "processing" as asked by the OP, which, as I interpret it, means the Develop Module. But I have my catalog file and my Lightroom software and my operating system on a single SSD drive, and I do not notice any particular speed issues on my computer. Maybe import and export would be faster with separate disks, but I don't see separate disks impacting anything else.