If Britain had been invaded (assuming the Germans had the logistics to do so), it's likely the USA would have remained neutral. We need to remember that the US was still being wracked by the Depression and many were against involvement in another European war, especially after World War I (the "Great War").
FDR's personal feelings wouldn't result in a declaration of war, as that has historically been up to the US Congress (the President has had the power to intervene in other nation's affairs since the Korean War, WWII is the last war in which the US issued a formal declaration of war). Given the layout of the country in 1939-1940, I'd say he would have an uphill battle to secure the declaration, even if the UK was invaded.
Only a severe tragedy/attack on American soil would have united the nation (and Congress) and led to war.
Of course, it's doubtful the Germans would have tried to invade Britain with ground troops in the first place.
On the matter of the Royal Navy, it's important to realize it was bypassed at the Jutland by German air forces during the invasion of Norway, following the landing of ground troops there via the sea. It was possible to avoid the Royal Navy, and the idea that the entirety of Eastern/Southern Britain could be protected from air strikes is implausible. Of course, the German planes couldn't fly off far from the Norwegian coast, so Britain was spared that front.