There does seem to be a popular perception that the RAF were on the brink of defeat in the Battle of Britain, and only saved when the Luftwaffe switched its focus from airfields to large scale attacks on London in September. There is a nub of truth to this; Fighter Command losses were outstripping replacements in late August/early September, and training had been cut back so replacements were scarcely combat-ready. Hugh Dowding was concerned that Fighter Command were going downhill, and instituted a system by which squadrons were categorised as Class A (a full complement of combat-ready pilots), Class B (up to six non-operational pilots) or Class C (at least three fully operational pilots), the latter being based in the North and Scotland where new pilots could gain more flying experience before being thrown into combat.

Both British and German intelligence was faulty, though; the British overestimated the size of the Luftwaffe, so were geared for a drawn-out war of attrition. The Germans underestimated the size of the RAF; both sides overestimated the number of kills scored, reinforcing the German idea that Fighter Command was all but wiped out by September. According to Richard Overy's The Battle of Britain: Myth and Reality Goering was informed in early September that Fighter Command had been reduced to a strength of 100 serviceable fighters; on 6th September they actually had 738 operational aircraft, with a further 256 in stores ready for despatch; Ken Delve's Fighter Command: 1936-1968 has a figure of 944 aircraft, 710 of which were serviceable on September 5th.

Something that's often overlooked is that though the Luftwaffe's overall size was considerably larger it was the fighter squadrons, particularly Bf 109 squadrons, that were critical to establish air superiority, and numbers there were more even (a strength of 1,107 single engine fighters in June 1940 according to Williamson Murray's Strategy for Defeat). The pilots were under even more strain than the RAF's with lower percentages of operational pilots and little rotation or leave. Fighter Command pilots had 48 hours of leave every two weeks, and squadrons served for an average of three weeks in 11 Group, bearing the brunt of the fighting in the South East, before being rotated to one of the quieter groups for a period. Apparently Luftwaffe veterans of the Battle would ask to see each other's appendectomy scars, a case of "appendicitis" being one of the few ways of getting away from combat for a time.

The Luftwaffe's switch to bombing London did indeed give Dowding some respite, but even prior to that the situation was not critical. Grass airfields were difficult to put out of action long-term; from 8th August to 10th September thirteen 11 Group airfields were attacked over 40 times in total, but only two were unfit for flying for more than a few hours. Luftwaffe intelligence failed to identify the crucial sector stations that controlled interceptions, even if they had been pinpointed most bombing was not accurate enough to hit precise targets.

There were actions on either side that could have affected the Battle. Leigh-Mallory of 12 Group favoured "Big Wings", assembling large formations of several squadrons to hit the Luftwaffe en masse, reasoning it was better to inflict heavy casualties on bombers even if it meant hitting them on the way back from their mission. Park of 11 Group sent his fighters in piecemeal, seeking to peel away fighter escorts and disrupt the bombers before they could reach their target. Park's tactics worked, part of the reason that Goering insisted his fighters should closely escort bombers, despite this being an inefficient tactic; if Leigh-Mallory had his way German bombing could have been more effective and the "Big Wing" would have been a prime target for free-hunting Bf 109s (Wood & Dempster, The Narrow Margin). Stephen Bungay in Most Dangerous Enemy highlights the effectiveness of an experimental Bf 110 fighter-bomber group, ErPro 210, who carried out accurate low-level attacks, and posits that if the Germans had fully appreciated the importance of radar and sector stations then a combination of commando raids and fighter-bomber attacks could have suppressed coastal radar stations, and allowed regular determined attacks on airfields to potentially drive 11 Group out of the South East.

Further speculation gets quite deeply into "what-if" territory (e.g. would an invasion have been launched if Fighter Command were forced to withdraw to more distant airfields), it's a subject that crops up quite frequently on forums like alternatehistory.com; without fairly fundamental changes on both sides it's hard to see a radically different outcome of the Battle of Britain.