British Standard Cycle threads use a 60 degree threadform, with a shape which visually resembles the Whitworth form. The 60 degrees makes the thread 'flatter', but the crests are rounded, like the Whitform threads, although the radius is different. The height of the BS Cycle thread from crest to crest is 0.5376p as opposed to the Whitworth at 0.640327p.The radius of the crest is p/6 (= 0.167p), as opposed to the Whitworth crest radius of 0.137329p.
So; BS Cycle threads are not Whitworth threads.
BUT, the author may have taken account of the fact that BS 811 : 1950 (still the current standard) provides the following note:
'It is customary practice in the cycle industry to use a 20tpi series of Whitworth form threads as an alternative to the cycle form thread series…'
That's not the same as saying that the BSCycle threads are Whitworth; and the practice seems only to apply to 20tpi threads.
Although, in general, BSCycle threads are 26tpi, the smaller diameters (1/8, 5/32 and 3/16 inch) are not. 1/8" is 40 tpi, while 5/32 and 3/16" are 32tpi. It would not surprise me to learn that the 1/8" BSCycle thread is sometimes used in place of the 1/8 x 40tpi imperial Model Engineer thread, especially where the focus is on the diameter and 40tpi specification in a mixed bunch of older taps or dies, but it is not the same thread, and should not be a fit for the proper Whitworth form ME thread. Mating the two will result in damage to the thread flanks even in a short hole.
Marcus