As is my understanding that anyone involved in the French Government at the time of invasion was either killed or fleed the country.
So after it was liberated, how was it decided who would run the country?
Surely if all the French politicians were killed, and that left the common man. So they would have no formal education on how to run a government.
Enlighten me, would ya?
As is my understanding that anyone involved in the French Government at the time of invasion was either killed or fleed the country.
False. Only 27 parliamentarians "Fled" aboard Massillia to continue the rule of the 3rd french republic from Algier, via Tanger. 649 parliamentarians voted the bill to give Petain the full powers (Full powers had been given before within republican times then went back to democracy) and only 80 voted against it (some 61 of the communist parliamentarians had been denied their voting right beforehand, in January 1940).
Aboard the Massillia, Jean Zay (Jewish, Minister of Education), Pierre Mendes France (Jewish, Deputy Minister of Finances), Édouard Daladier (National Defense and War Minister and ex-President of the Council), Georges Mandel (Minister of the Interior) and César Campinchi (Minister of the Admiralty) where heading toward Tanger to lead the war from there. De Gaulle (Deputy Minister of National Defense and War) was their "invoice" to London. By the time the Massillia ship boarded Tanger (modern-day Morocco, then a French protectorate) Petain had been given power and the armistice was signed. Most of the parliamentarians where arrested as Traitors and/or Deserters and/or Jewish. Jean Zay was latter executed in his cell as France was being liberated and Piere Mendes-France went on to being a prime minister of the 4th French Republic (then the post who held the most power).
In 1943 the "National Council of the Resistance" was founded by De Gaulle as the only free man left of the elected french government that never recognized the armistice/defeat. The National Council of the Resistance regrouped most of the french forces,fighting and political ones, against the Axis and the Vichy regime. It maneuvered to unite the french resistance (from right-wing nationalist to Anarchist and a very strong communist movement) under their banner and to get international recognition. Their strong communist base (while legitimate, the communist party would become the 1st in electoral vote in the next elections) handicapped them in getting recognition from the Allies but De Gaulle, being a right-wing politician was able to maneuver and get recognized as the leader of Free France. After the war this body drafted the constitution of the 4th French republic, leading De Gaulle to leave the political scene as he viewed this constitution as leading to too unstable governments. He came back in 1958 to draft (well, mostly Debre but he was in his political party and De Gaule asked for the re-writing) the 5th French republic, which is (and became even more) a presidential system, and it still in application today.
All my sources are in French but here is a few wikipedia articles in English to give you a start:
Note: some exact numbers are from wikipedia.
EDIT: reorganized the information and added some more.