Some parts of the French Navy did join the Allies. Lieutenant de Vaisseau François Drogou and a band of volunteers, took the French submarine Narval into Malta on the 26th June 1940. The captain and crew of the French submarine Rubis, based at the British port of Dundee, volunteered to join the British a few days later. Other officers were unwilling to join the British, but found compromises that suited both parties; Vice-amiral Godfroy, commanding the French squadron at Alexandria, allowed the British to intern his squadron there. However, the majority of the French Fleet was unwilling to join the Allies. The reasons for this were multiple: German threats, complex interpersonal and national loyalties, and poor British diplomacy all played their parts.

The armistice between France and Germany was signed on the 22nd June 1940, near the small town of Compiegne (where the armistice that ended WWI had also been signed). Of the armistice's 24 articles, Article 8 concerned the French Navy. It stated that

The French Fleet (with the exception of that part which is left at the disposition of the French Government for the protection of French interests in the colonial empire) will be concentrated in ports to be determined and will be demobilised and disarmed under the supervision of Germany or, respectively, Italy. The peace-time home bases of these vessels will be used to designate these ports.

In other words, the French Navy would have to disarm in French ports under the supervision of the Axis powers. Article 8 also contained assurances that the French Navy would retain control of the disarmed ships, and that the Germans would make no attempt to take control of them or use them for military purposes. On the face of things, this should have been acceptable to the British too, but they did not trust the Germans. In effect, Article 8 placed the French Navy at the disposal of the Germans, and if they chose to break the terms of the armistice and sieze it, there was little the French could do. To the French, however, it was perfectly acceptable. Equally important to understanding French actions was Article 10. This stated that anyone who left France to fight against Germany would be considered a franc-tireur (essentially, an person unprotected by the laws of war), and forbade the French government transferring any assets to the British. This was a powerful deterrent to anyone considering joining the British. The Germans had inflicted harsh reprisals on franc-tireurs during WWI, and it was clear that such policies would be repeated. While they might not be able to reach those in Britain or in the French West Indies, their families or property in France would be at risk. Only those who were devoted enough to the Allied cause to lose them would be willing to take the risk, and few in the French Navy were.

This was because most of the French Navy's officer corps was loyal to France, and to Admiral Darlan, head of the French Navy, not to an Allied cause that seemed to be on its last legs in June 1940. The swift defeat of France had thrown the Allies into a relative despair; it seemed likely that Britain would meet a similar fate. There was little enthusiasm in France for continued fighting. The idea that Germany was a unique evil and threat that must be defeated had not taken hold; this was a later development that grew out of German actions in the following years. There was also no clear leadership amongst the Free French forces in Britain; General de Gaulle had yet to take his place at the head of them. With no charismatic leadership in Britain, French Navy officers tended to stick with their prior loyalties. Many senior officers had close relationships to Amiral Darlan, head of the French Navy. Darlan had helped these men into their positions, had worked with them throughout their careers. He was also a strong supporter of the Vichy Government. In the absence of an equally influential officer in Britain the French Navy's senior officers looked to him, and stayed in France. This also let them stay loyal to France. The Vichy Government retained a continuity with the Third French Republic, and with no Free French Government, could easily present itself as the legitimate government of France. Darlan's instructions to the French Navy emphasised this. The French Fleet was not to be handed over to either the British or the Germans and Italians; it was French, not a playing piece to be used by either side. If either side tried to seize it, it would fight, scuttle itself, or move to a location that was out of reach.

However, there was a chance that diplomacy could still pry some of these officers away. As noted above, Vice-amiral Godfroy at Alexandria was willing to work with the British. This was the result of careful and tactful diplomacy by the British Admiral Cunningham, with whom Godfroy had a close working relationship. Similarly tactful treatment was given to the crew of the Rubis at Dundee. Unfortunately, the British often chose to take a more forceful tack. In Portsmouth and Plymouth, ships were boarded and seized by force. This was mostly done without violence, but aboard the submarine Surcouf, a gunfight broke out that left four dead (two British officers, a British seaman and a French NCO). At Mers-el-Kebir, negotiations broke down, due to pressure from the British government and a number of tactless choices by Admiral Somerville, the man on the spot; I've covered this in more detail in this thread. Following Mers-el-Kebir, there was little chance of convincing the French Navy to join up with the service that had killed hundreds of their colleagues.