On the extremely rare occasions (I can only think of one, although I admit that it was a very significant one) that Fascist regimes have come to power in countries with monarchies, then they have abolished them as soon as possible in order to get into their stride.
Leaving aside exactly what Franco was or was not - he was never a member of the Falange, never attended the whole of any of its conferences, and was thoroughly despised by the real thing in the Fuerza Nueva - it is undeniable that Spain broke with his legacy specifically by restoring the monarchy.
If numerous local kings, princes, grand dukes and the rest had still been doing what such figures do throughout German-speaking Europe and the Italian Peninsula, then Hitler and Mussolini would have been impossible, since there would simply have been no gap to fill. The ludicrous Mosley, a pioneering European federalist but whose importance at the time is wildly exaggerated in order to do the same for his Communist opponents, openly wanted to abolish all hereditary titles (not his only similarity with New Labour), although that never stopped him from using his own.
When he attends a Buckingham Palace Garden Party, Nick Griffin can doubtless explain all of this to the Queen as the background to his own party's policy of abolishing the monarchy. When not discussing Her Majesty's own descent, both from the "negroid" Charlotte of Mecklenburg-Strelitz, and, via Elizabeth of York, from Muhammad.
This really will be the last opportunity for either of them to have that conversation. The BNP took barely half as many votes this year as last year, fought 38 council seats and lost 36 of them, and lost every seat on Barking & Dagenham Council, of which it had thought that it might take control.
Like the NF before it and the BUF before that, the BNP was always talked up by its sectarian Left opponents, in order to make themselves appear more important than they really were. This remains very much the case with the teaching of the 1930s. Mosley was never really terribly important. Nor was Griffin.
Every thirty or forty years, this tendency reappears to make a lot of noise before going away again. Thankfully, I will be in my sixties or older before it next happens. And thankfully, it would, once more, only last for a few years. That is how these things work.