The challenge is to have the controversial Marshall Petain decide against the armistice and instead become the political leader of the Free French movement. There are a number of difficulties with this scenario mainly because Petain was the most prominent member of Premier Reynaud's Cabinet calling for armistice in OTL. If he is more stridently calling for continued resistance then it might embolden Reynaud to continue in office rather than resign. Reynaud resigned his position only after indicating to General Weygand he wanted a military surrender in metropolitan France, with the Government relocating to Algeria and taking command of the overseas military assets to continue the war. Weygand refused to do this which left Reynaud powerless. While Petain's change of heart in this world might prevent this, for the purposes of this scenario I'm assuming the armistice still goes ahead.
Petain was quite old by this point, so his role may be more symbolic than organisational, but even then his prestige and clout will lend a huge amount of credibility to the Free French movement. I'm not sure if it's strong enough to butterfly away American recognition of Vichy, but certainly I could see more then just Equatorial Africa pledging allegiance. There were also roughly 20,000 French soldiers in the UK having come back from the failed Norway Campaign, most of whom returned back to France. With Petain in charge instead of De Gaulle I think it likely a great deal more of these soldiers remain behind and enlist in the Free French forces, perhaps as many as 2/3 (De Gaulle was relatively unknown by this stage).
I acknowledge this will be difficult to pull off. Petain was famous for his disdain of the British, and he was privately remarked to have been a defeatist in the early stages of WW2 (and even allegedly WW1). The trick would be to make him hate the Germans more than the British and recognise the Nazi menace as disastrous for France.
I think the biggest change will be after the war, particularly politically. Petain was a very conservative man and will be blaming France's fall on the socialist government of Blum and to a lesser extent Daladier. Without Petain's fall from grace many will probably agree with him and the far-right movements (those that didn't outright cooperate with Germany at least) might receive considerable political support.