Hitler didn't meddle in strategy. The Luftwaffe commanders requested permission to attack London, which had been their plan for the end game all along. Hitler granted it. In particular, Kesselring had been pushing for the attack to switch to London for some time, Goering was also anxious to do so because his repeated boasts that the RAF would be destroyed in the next few days had all proven false.
Actually there were British reserves left, although they had dwindled since the start of the BoB. But front line strength, ie the number of aircraft in active squadrons, had actually increased.
The total number of serviceable Spits and Hurris in reserve went from 329 on 12th July to 224 on 6th September (the BoB in British terms began on 10th July, the Luftwaffe switched to attacking London on 7th September)
However, front line strength went from 736 Spits and Hurris on 12th July to 816 on 6th September (these are strength figures, not all would be immediately serviceable, although serviceability was high, and this doesn't include aircraft needing more than minor repair). In other words, over the most intense period of the battle, the total number of Spits and Hurricanes available declined by 25, out of more than 1,000 aircraft.
In contrast, the Luftwaffe expended their reserves very quickly, by September they had no aircraft in reserve, and front line strength had begun to decline sharply. The Luftwaffe had about 1120 109s at the end of June, by the end of September that had reduced to about 920.
As to range, the vast majority of the battle was fought within a hundred miles of the 109 bases. Indeed, the complaint that 109s only had 15 minutes combat time over London should be viewed with two things in mind. First, that the USAAF typically allowed 15 - 20 minutes combat time for their escorts over Germany (the USN allowed 20 minutes, iirc) and second, that by the time the battle reached London, the Luftwaffe had already lost. They'd lost by failing to protect the bombers, and suffering far too high a casualty rate, in battles against the RAF at very short range, typically 50 - 75 miles.
The Luftwaffe didn't lose because they couldn't bomb targets in the Midlands. Their goal was to defeat the RAF, and they attempted to do so with an all out assault on 11 Group. Longer ranged fighters would have meant a diversion of effort, and brought 10 and 12 Groups into the battle as well.
If you can't win when you concentrate all your force against a small portion of the enemy force, then you won't do better by spreading your attack out. That would have given 11 Group a respite, meant deeper penetration raids for the Luftwaffe, which would have given the RAF more chances to intercept.
The only way longer ranged fighters for the Luftwaffe would have helped would have been if the RAF had withdrawn it's fighters north of London (in the event they won anyway, and didn't need to withdraw) or if the British didn't have vital targets in the SE they needed to defend. As it was, the SE contained the most vital targets in Britain.
It was actually losing too many of both. Milch did a tour of Luftwaffe bases in France in late August and early September, at the height of the battle. He found that the gruppen were seriously under strength, with many having only half the aircraft they were supposed to have, and half as many fit pilots.
Why does the Luftwaffe have to defend airfields in France? The RAF had to defend fields in Britain because they had to defend British airspace, and needed the airfields to do so. The Luftwaffe have no need to defend French airspace, so no need of airfields. Even though they historically kept fighters in France, there were far fewer fighters spread out over a much larger area, making attacks on airfields much harder for the RAF in 1941 than for the Luftwaffe in 1940.
