The information you want is here:
British Air Defenses of England, 1 July 1940
usacac.army.mil/cac2/CGSC/CARL/nafziger/940BGAA.pdf
The Nafziger collection of Orders of Battle at the US Army Combined Arms Research Library (CARL) is a godsend for any game designer, wargamer, or student of military history.
The only limitations are that most of the orders of battle don't give exact troops strengths, nor do they get into details as to exact models of equipment on hand (e.g., Spitfire IA rather than just "Spitfire"). Likewise, they don't discuss "human factors" such as morale or training, nor logistical issues such as state of supply, spares on hand, or serviceability percentages.
For that sort of really geeky stuff you have to dig into the wargamer community. There have been guys studying this stuff in intense detail since the 1960s at least. Old tabletop wargames and specialist magazines often give useful detail that general histories, and even many of the specialist publications aimed at modellers, miss.
That said, the list of RAF Squadrons actually looks pretty good. The fact that it includes details that are useful for a game designer like squadron codes and call signs sets it apart.