No Italian aircraft or aircrews were used by the axis in the battle of Britain it was entirely a Luftwaffe affair. The Germans were always contemptuous of their Italian allies with their outdated aircraft and poorly trained personnel.
The battle at El Alamein was fought between German forces (Axis) and British. The outcome of the battle (a British victory) was a turning point in the war. It was said after the war, that before El Alamein, Britain never won a battle and after El Alamein it never lost one.
Britain was attacked by the Axis powers, specifically Germany, during the Battle of Britain. It was an air battle fought mainly over the Southern counties of England. The German purpose of starting the Battle of Britain was to defect British Air Superiority as a pre-cursor to a land invasion.