No he wasn't a serious threat to British democracy because the same conditions that facilitated Fascism and Nazism didn’t exist in Britain. Even though the war was a trauma, Britain definitely came out a winner, so you didn’t have legions of enraged nationalist veterans feeling cheated (as in Italy and Germany, albeit for different reasons), the Communist Party of Great Britain was growing but it wasn’t a serious threat while the first Labour government of Ramsay Macdonald turned out to be pretty moderate (probably too moderate for its own good). Also, Britain had indeed lost territory after the First World War, Ireland, but the loss of the 26 counties didn’t feel like a real loss largely because the Irish question had been such a headache in British politics and the Irish Free State was still part of the Empire (whose end was still decades off).

All of those factors (resentment from WWI, irredentism or desire for territory and elite fear of a rising Far Left) had been instrumental in the rise of Mussolini and later Hitler. The editor of the right of centre, Daily Mail did briefly endorse Mosley with an editorial entitled “Horray for the Blackshirts,” but once he realised that British Union of Fascists were not just some eccentric, patriotic, conservative pressure group he withdrew his support. It should also be noted that both the United Kingdom and France had a much longer history of democracy and constitutional government than Germany or Italy, ergo totalitarianism didn’t feel natural to most Brits.

That’s not to say that the BUF didn’t attract a somewhat noticeable base of support; it was far more significant than the smaller British fascist parties that had proceeded it (the British Fascisti and Imperial Fascist League) or that Mosley wasn’t a talented politician in many respects. Had he remained a Tory or Labourite, who knows? He was considered handsome, well spoken and charismatic.. That can be a very potent combination in any country’s politics.

The reasons for their limited success were of course Mosley’s notoriety, money and political skills, and probably the fact that initially Mosley modelled his movement off of Italy’s National Fascist Party rather than the National Socialist German Workers’ Party. The other British fascist movements to date had been your typical suspects on the Far Right (raving anti-Semitic conspiracy theorists and thugs), while at first Sir Oswald Mosley and his British Union of Fascists did at least appear to be different.

In a very counterfactual universe in which Britain had been on the losing side of WWI and was humiliated by Germany, sure “maybe” Mosley could’ve taken power, but otherwise no way. That’s not to say that Mosley never could have become prime minister, but he would had to remain in the Conservative or Labour Party for that to have been theoretical plausible.