It's a fine balancing act.
Initially one might suspect that the damage inflicted upon the bombers would be greatly increased.
In November 1939 Dowding wrote that he estimated as many as 300 .303 strikes would be required to bring down an He 111. About a year later he received an anonymous letter (dated October 3rd 1940) via the RAF Staff College at Andover. It quoted the opinion of an anonymous "test pilot who was attached to a squadron and bagged four huns." The opinion would have reinforced what Dowding already knew, that the existing armament was good for destroying fighters but "is rapidly becoming obsolete for use against enemy [bomber] formations."
The maths doesn't make happy reading. A two second burst from eight .303 machine guns will deliver about 300-320 rounds down range (assuming about 1100-1200 r/min for each gun). If Dowding's estimate of 300 to down a bomber is even close then a fantastically high standard of marksmanship would be required to shoot down a bomber in one firing pass, and one pass is all the attacking pilot might get.
There is no doubt that a 20mm cannon would have inflicted more damage on the Luftwaffe's bombers. The fundamental problem that the change in armament cannot address was the inability of most pilots to hit anything, whatever weapons they were equipped with.
RAF scoring can be misleading. The system of giving fractional scores means that a substantial percentage of pilots (nearly 40%) have at least some sort of credit, two halves appear to be one whole in the statistics, but only about 15% claimed a whole aircraft in one go so to speak.
221 of the Luftwaffe aircraft shot down in aerial combat during the battle were shot down by just seventeen pilots.
The competent pilots would surely have scored more highly if armed with cannon, but the rest would probably have done no better. The damage inflicted on Goering's precious bombers would have been increased, but maybe not as dramatically as it might first appear.
Cheers
Steve