Pre May 1962, there isn't really a steady pattern, as all necks were then made by hand, without any A/B/C/D designation, so some were more chunkier than others. There are some patterns in it, but they're nowhere conclusive or steady. Sometimes it was a slim C-shape profile, then the next day it could be a chunky D-shape, or even U-shape. The necks from 1959 (when the rosewood fretboard was introduced around August) are chunky U-shapes. They became thinner C-shapes in 1960 & 1961, before going to chunky D-shapes in the early months of 1962.
After May 1962, a standardisation happened, with A, B, C & D coding. A was the narrowest at 1.5" and was the standard for the Jazz Bass. B was next at 1,625", and was the standard for the Precision bass. C was 1.75", and D was 1.875". For a P-bass, A, C & D were available, but you needed to order them from the factory. If you didn't specify the neck width, you'd get a B-neck. B & C necks are equally common (B slightly more common), A is uncommon and D is very uncommon (D was more found on guitars).
As for thickness (front to back). Pre-CBS was usually the thinnest, with some very fine necks being made between 1962 & 1964 (they might by wide, but also very thin), after 1965 they thickened a bit. But then again, while the widths were now fixed with routing templates and a numbering system, the thicknesses weren't, so you could fine thicker and thinner necks from the same year, as they were all handmade.
Brazilian Rosewood was phased out at the end of 1965, with sometimes old-stock being used up later on (someone I know as a '72 P bass with a fretboard that really looks like Brazilian).
For the body: from 1959, Alder was the wood of choice for every finish, except "white blonde", those remained Ash (for the reason that ash had a distinct grain pattern that could be stained, and would show nicely under the semi-opaque blonde finish). But those are rare, as "white blonde" had to be custom ordered. In 97% of the cases, you're dealing with alder bodies in the 1960's.
For your wishes, a slab-fretboard with a narrow(er) nut and thinner neck, a 1960-1961 P-bass would tick all the boxes.