A bit of background may help.
When the industrial revolution started there were no standard threads – everyone did their own thing, so nothing fitted together unless made in the same workshop to do so. This caused lots of trouble!
Whitworth realised the value of standardising threads for fasteners. As he was big into measuring, he collected as many nuts and bolts as he could from British industry and averaged their dimensions. The resulting form, 55° rounded top and bottom, gradually displaced most other UK threads during the early Victorian period, and, when the British Standard Institute was created, they made it 100% respectable.
Whitworth isn’t ideal for everything! The form is well adapted to soft materials like Cast Iron, less well to steel, which appeared 30 years later. The original standard only covered coarse threads, so fine threads were added later. The rounded form adds strength, useful in Whitworth’s day, but expensive to make. In the USA Sellar’s later thread switched to 60° with a flattened form – cheaper. Likewise, on the Continent, for the same reasons, designers went for 60° and a flattened form, but based on metric pitch rather than turns per inch. Later threads were designed from theory, rather than Whitworth’s averaging process. Whitworth is still good for soft materials.
By 1870 another problem with Whitworth had emerged. The thread came from heavy engineering and isn’t well-matched to light-engineering requirements: clocks, small-arms, scientific instruments, gas meters, and especially electrical. They call for a different thread form, so the British Association for the Advancement of Science tweaked a Swiss thread to meet the need. So BA has a 47.5° thread, with a metric pitch, and Imperial dimensions. BA was extremely successful, used whenever a fastener of less than 1/4″ inch diameter was needed. Very common in older equipment of all types.
Later, it was found that the standard metric series in coarse, fine, or extra-fine got equivalent performance to BA and were cheaper and more acceptable to foreign customers. Thus BA gradually withered.
In the 1/4″ cross-over region older British equipment could be BSW, BSF or BA depending on what the designer thought best.
After WW2 Unified Threads arrived causing even more variation. One car I owned in the 70’s had UNC, UNF, BSF, BSW, BA, and metric fasteners. Made by a company created by merging US, British and continental car makers, each of whom contributed parts …
Dave