There's a really useful post by u/Vespertine on the popularity of the BUF, especially compared to other political parties of the time. This thread might also be useful in terms of understanding the popularity of the BUF over time, and the extent it was effectively opposed. Essentially, even at the BUF's peak in 1934, it commanded neglibible electoral support and its membership was dwarfed by that of mainstream parties. Britain was never particularly 'fascist' in the 1930s.

However, one caveat I would like to make about this scepticism is that the tactics and nature of British fascism makes such a portrayal as you describe in the ABC Murders a bit more plausible (to be clear, I've not seen it, and can't judge it directly). The BUF was not playing a typical political game by the standards of the time. Mainstream parties such as Labour or the Conservatives were geared to fight and win elections, not control or contest public space. In contrast, smaller, more extreme fringe groups such as the BUF (and Communist Party), who had very limited electoral prospects, utilised tactics that emphasised constant, focused campaigns, and maintaining a visible, voluble public presence. By holding mass rallies, contesting control over public space and creating bold political pageantry, these parties attempted to show themselves as being at the head of a popular, powerful mass movement. Their success was measured chiefly by the ability to mobilise support on the streets, not at the ballot box.

In this sense, it is not out of the question that in certain parts of Britain (notably places such as London's East End, targeted for repeated campaigns by the BUF in the mid-late 1930s), fascism might have a relatively constant, visible presence. However, so did their opponents, who were ultimately more successful in mobilising mass opposition to the BUF. Moreover, sympathies shifted markedly over the course of the decade. In the early 1930s, it might not be uncommon for middle-of-the-road types to have some sympathy for fascism as practiced in Italy under Mussolini, or Germany under Hitler - these were clearly inferior countries, with big problems, who needed a strong, firm hand to sort them out. But this was not at all the same as thinking that fascism was appropriate for Britain, and by the late 1930s the true nature of fascism was much more apparent to the bulk of the population, reducing even this kind of casual pro-fascist sympathy to very low levels.