Bustr makes the best point so far and I will add to it;
First off the British did not possess a licence to produce the US 12.7mm Browning HB machine gun. It also had no factories to make it even if a licence would have been forthcoming from the Americans.
May 10th 1940 the Germans invade France. By late June its apparent that France is lost, and the Luftwaffe moves its bases closer to Britain. July 1940, its apparent that Britain may face invasion and an air campaign. Lets say at that point the RAF gets all its ducks in a row and decides to go with the 12.7mm, convince the Air Ministry that its needed, and gets govt approval to get the new weapon. OK. Tall order but ok, lets just say...
The British at that point have to go the United States and request several thousand 12.7mm machine guns and @ 1 million rounds of ammo. First off, the US Congress has to approve it, which takes time to do. They also might have said no. The USA was not at war with Germany at the time and only certain war materials were being shipped to Britain. Lets say they said yes. Next you need to put an order in with Browning, and have the guns produced. Next gather them up, ship them to the east coast, load them into a convoy (more like several convoys) and sail it to Britain. At the same time you need to redesign the Hurricane and Spitfire, neither of which is designed to take 12.7mms. When the guns arrive in Britain, you send them to the factories producing or refitting the fighters and install them and then start sending the new fighters to front line units.
...and do it all before August 1940 when Eagle Day is launched and hope that there was no delay in fighter production. Unlikely is an understatement.