For poultry and game, the name of the meat is usually the same as the name of the animal (chicken, turkey, rabbit, pheasant, grouse, etc.). The only exception I can think of in this category is venison (derived from the Latin venationem, meaning hunting).
The same is true for fish and seafood.
For farmyard animals, “lamb” describes both the creature and the meat. That leaves veal, pork, beef and mutton (and any others I haven’t thought of). Of these, the OED records that all except beef are or were also used to describe the living creature, viz:
“Hogs and porks, the word appearing to be used indifferently, are occasionally found.” (1887)
“My mother…would receive her prodigal and kill the fatted veal for me.” (1855)
“The word mutton is sometimes used [in America], as it once was in England, to signify a sheep.” (1833)
Beef, the odd one out, seems to have been used to refer to oxen but not cows:
“Behind these came a beef, driven by soldiers… The beef was immediately shot at and butchered.” (1878)
I think that this, basically, supports what the previous posters have said: we use the French word when it’s food and the Anglo-Saxon word when it’s walking around.