While fascists movement did exist and did rise in France, they were never able to take power there though the infiltration of the political mainstream with some of their thoughts and ideas did nonetheless contribute both to France's quick defeat in WWII and the subsequent collaborationist regime in Vichy France.
The various reasons why fascist movements rose and were able to take power in various countries in the inter-war period are manifold and the result of a vast variety of contributing historical factors, however, shared characteristics that much of the literature points to as crucial are the loss of legitimacy of the revolutionary post-war order, civil-war or civil-war-esque conditions, and the presence of scores of radicalized young people – in particular men – who took the mindset that developed during the war and was so prevalent as basis for a new political ideology.
The impact of WWI can not be understated when it comes to the political shape of the 20th century. The end of WWI truly brought the end of the long 19th century and began what Eric Hobsbawm termed "the Age of Extremes": Millions of people dead, countries exhausted to the brink of total destruction and a variety of political systems in shambles. Added to that the Spanish Flue, which killed more people than the war itself and a total change of most of the political map of Europe accompanied by Civil War and massive brutality. With all these factors in play, it is not surprising that whatever system had been in charge would be de-legtimized and collapse under public pressure. Hence, the end of WWI lead in most countries to a political revolution – Germany, Austria, Hungary, Yugoslavia or in a delayed way Italy etc. – if not a social one or an attempted social one – the Soviet Union, Germany, Austria, Hungary etc.
With these revolution and the change they brought in the political arena came in the countries most affected a feeling among the middle class that everything they had been brought up to believe in and value – the state, the monarchy, saving money – had completely been devaluated. Describing post-war Germany to build upon his description of the later leaders of the Nazi Reich Security Main Office, Michael Wildt paints a picture of post-war Germany in which young people who profiteered from the war were spending money from the black market as fast as they could make it in Berlin clubs on Champagne, while the traditional middle class of the German Empire had lost all their savings investing in war bonds and through the self-inflicted inflation resulting from the Ruhr Boycott. The same time saw Social Democrats in power and Communists trying to instigate revolution in both Berlin and Munich thus solidifying further the fear of descent into Bolshevism, losing everything and the loss of faith in everything that had been valued before.
And while France too was economically on a human level devastated by the war, the crucial difference was that its political system – despite the massive assault it would come under in post-war years from left as well as right – was able to stave off destabilization by selling the war as a victory – maybe a hollow one but a victory nonetheless. At the same time, Britain was helped massively by control over its colonies on both a manpower and economic level. While far from the problems of Germany, Italy or France, it too however had to make some concessions such as the independence of Ireland that followed the war closely.
Additionally, one cannot underestimate the mindset that developed from the last phase of the war 1917-1923 and that was crucial for the appearance of Fascism in the first place because it shaped and crafted the key attributes of Fascism as an ideology: Conflict between races driving history, a utopia in form of glorified past that never existed, a characterization as a Weltanschauung, and a victimization myth that could only be rectified through redemptive violence.
In his book The Vanquished. Why the First World War failed to end, 1917-1923 Rober Gerwarth writes:
It was in this period [1917 forward] that a particularly deadly but ultimately conventional conflict between state – the First World War – gave way to an interconnected series of conflicts whose logic and purpose was much more dangerous. Unlike World War I, which was fought with forcing the enemy to accept certain conditions of peace (however severe), the violence after 1917-18 was infinitely more ungovernable. These were existential [sic] conflicts fought to annihilate the enemy, be they ethnic or class enemies – a genocidal logic that would subsequently become dominant in much of Europe between 1939 and 1945.
In countries such as Germany, Austria and Hungary that were much more effected by this violence than France or even Britain this lead to a formulation of fascists ideas and them festering, especially in the milieu of conservative radical revolution that was in search of new utopia besides monarchy and a generally radicalized generation of people who throughout their adult life had never experienced the time before the war and were completely ideologically engulfed in the war's political radicalism and hatred.
However, the mere presence of these ideas did not lead to their rise to power alone – as I mentioned above, France too experienced the presence of a whole variety of fascist movements without descending into fascist rule. While in Italy the rise of Fascism was precipitated by fear of communist takeover, a crisis of legitimacy of the political system, and a massive irredentism based on little gains despite nominal victory in the war, in other countries it took additional events that lead to the crisis of legitimacy of the established system: In Germany, the world-wide financial crisis of 1929, in Austria, a massively partisan political system, in Hungary, the counter-revolution against the communists and so forth.
As many scholars point out, it is not just crisis of the economic variety but yet another key factor: The reaction of political and social elites, meaning by en large, the conservative political movement. In countries such as France and Britain where there was a long established tradition of a democratic right and a democratic conservatism, the political system was able to weather various crises. In Germany, Austria, and Italy where there was not a long tradition of a democratic right-wing in politics, it were these social and political elites of the right – the king, the bureaucracy etc. – that enable fascism to rise and take power. Where traditional right-wing and conservative classes did not believe in democracy and the left found itself constrained by Moscow and its own commitment to compromise with the anti-democratic right, Fascism was able to rise and take power. This was specifically the case in Germany, Austria, Italy, while it was not so in France and especially not in Britain.
So, to sum up: The historical factors we can identify as especially pertinent for the rise of Fascism in Italy, Germany, and Austria and that France and Britain lacked in comparison were:
political revolution and attempts at social revolution at the end of WWI
a radicalized political ideology based on a mindset fostered in the war and a middle class which had experienced a complete loss of faith in what they regarded as most important
a left that was constrained by either Moscow or their commitment to compromise.
a brutalization of political discourse shaped by civil-war and the most violent conflicts within WWI 1917-1923
a conservative elite and right-wing political spectrum that lacked a traditional commitment to democracy and liberal (as in classical European liberal) order
and a further crisis of political legitimacy of the post-war political system brought about by the economic crisis of 1929, a political discourse that devolved into violence or a civil war based on ideology
There is more to be said about individual cases but this about sums it up in a very general way.
Sources:
Robert Paxton: The Anatomy of Fascism.
George L. Mosse: The Fascist Revolution: Toward a General Theory of Fascism.
Robert Gerwarth: The Vanquished.