The single event that put a dent in the BUF’s power and propaganda was the end of its access to the press. The Daily Mail and Daily Mirror were its main propaganda tools. Their owner, Lord Rothermere, stopped supporting Mosley after the fascists were accused of initiating brutal violence during a meeting at Olympia in 1934. After that meeting, Rothermere’s Jewish advertisers in the UK threatened to pull their advertising unless he stopped editorially supporting Mosley.  Without the press, the BUF’s message was limited, and its membership dropped to 5,000 the following year. The final nail in the BUF’s coffin came in 1940, when the government banned them after the start of WW2.
So, the lessons to draw from Cable Street and the other anti-fascist actions in the 1930s are:
1) Violence is not an effective long-term tactic against Nazi hate groups. When Mosley’s fascists were perceived to be the victims of violence, their membership grew; but when they were perceived to be the perpetrators of violence, it dropped.
2) What does work, but is more difficult for peace groups to achieve, is applying economic pressure to the fascists’ financial base and swamping their propaganda with truth. This requires a long-term organizing strategy beyond the occasional demonstration or peace march (a good example of a long-term nonviolent strategy is the BDS movement).