In the UK women played an extremely important, and also extremely undervalued, role in the air war as pilots of the ATA - Air Transport Auxiliary. The ATA was a civilian formation whose purpose was to ferry new, repaired and even damaged aircraft to and from airfields across the UK to every destination (including front line airfields, factories and scrap yards - but excluding aircraft carrier). It also flew personnel on urgent assignment to their destinations and conducted air ambulance duties where appropriate.
Some 168 (according to the BBC, but wikipedia says 166) women flew for the ATA throughout the war under Commander Pauline Gower, MBE and all were volunteers from across the commonwealth as well as pilots from the USA, the Neatherlands, Poland and one from Argentina. These women were, at first, limited to flying non-combat aircraft but were eventually permitted to fly all aircraft - from Spitfires to flying boats and Lancasters.
I don't know if any of the female ATA pilots were involved in any dogfights when ferrying fighters (even if they were, I'm not sure they would have had any ammunition) but seeing as it is known that female pilots flew Spitfires from the Supermarine factory in Hamble, near Southampton, it's possible. The BBC article linked above says that 15 women died while ferrying aircraft.
Pilots in the ATA included the pioneering British aviator Amy Johnson, who died along with 14 other female pilots while undertaking the vital role of ferrying military aircraft as part of the war effort.
As an interesting side note, the women of the ATA were the first women in the UK to be paid the same as a man doing the same job from 1943, which was the first time the UK Government authorised the practice.
It's my personal opinion that these women played a very important role in the air war fought from the UK - without their efforts front line pilots would have been without aircraft for longer and would have had to ferry aircraft themselves. The pilots of the ATA undoubtedly played an important role in the Battle of Britain (although, it is impossible to say what would have happened without them), and through their duties as air ambulance pilots would have helped to save many, many lives.
There are plenty of books available about women flying in the ATA (I haven't read them all mind):
Spitfire Girls - Gould, Carol
The Forgotten Pilots - Curtis, Lettice
The Female Few: Spitfire Heroines of the Air Transport Auxiliary - Hyams, Jacky
Womens Auxiliary Air force (WAAF)
Another formation of women that were vital to the air war in the UK, and as a result the ETO, was the women of the WAAF. At it's peak up to 2000 women a week were enlisting into a force that numbered in excess to 180,000.
These women performed duties that were wide ranging and extremely important. While the only female pilots were limited to flying in the ATA, the women of the WAAF crewed barrage balloons, worked on meteorological and intelligence reports, aircraft maintenance, transport and communications.
To highlight the role that these women played, and their commitment to it I have this quote from the WAAF association website:
Looking at the Memorial, those of us who were there reflected on our past, when so many of us were plucked from our quiet lives by war. We travelled from towns, cities and villages in these islands and from countries further afield, making our way to recruiting centres to offer our services to our country. We didn't know what to expect and for many it was a rude awakening: the early mornings, the chores, the discipline, and for many the frightening crash of bombs and roar of aircraft, most of which weren't ours.
Our lives changed dramatically but in that maelstrom of war many of us
grew up, learned to share, discovered comradeship and learned new
skills. We learned to cope with death and disaster on those airfields
so long ago. On bomber and fighter stations we lost many friends,
mostly men, but bombs are indiscriminate and WAAF suffered too. As we
left the memorial, with its long lists of trades, we remembered our
friends from long ago, remembered them with affection and for some,
with sadness. We will always remember them.
Women Airforce Service Pilots (WASP)
The Women Airforce Service Pilots was the result of combining the Women’s Flying Training Detachment and the Women’s Auxiliary Ferrying Squadron (both formed separately in 1942) in 1943. While WASP came under the direction of the US Army Air Force, they were strictly civilian pilots and were not according any military benefits or honours. Unlike the women of the ATA, the WASPs were also paid less than their male counterparts. Despite this, the women were trained alongside USAAF pilots at Avenger Field in Sweetwater, Texas. WASP actually has some very impressive figures, some 25,000 women applied to join the service but only 1,830 were accepted and 1,074 passed out of training. These women all held commercial pilots licences (the ones who were accepted anyway) and apparently flew around 60 million miles.
We were still civilians. All of our training was to make [Army] Air
*Jane Tedeschi (CNN Article)
Much like the ATA, the WASP was a paramilitary organisation designed to free up male ferry pilots to take up combat duties. The women of the WASP flew almost every type of US Combat aircraft and were the first women to do so. Their primary role was to fly the aircraft from their respective factories to USAAF training bases and ports of embarkation, alongside this they also flew transport flights, towed AA targets and simulated strafing runs – among other things.
While the women of the ATA and WAAF received recognition throughout and after the war the women serving in WASP were not even accorded basic honours. Some 38 women died, all in accidents, while on duty and their families were expected to pay for their remains to be returned to them and the flag of the United States was not allowed to be placed on the coffin. After the war, the women did not benefit from the GI bill until the GI Bill Improvement Act of 1977 – in fact their role was not fully uncovered until earlier that year.
After the program disbanded, WASP records were sealed, marked secret and stored in archives for more than 30 years, Parrish said. The records were unsealed in 1977 after an Air Force press release that erroneously stated it was training the first women to fly military aircraft for the U.S. The women who flew military airplanes during World War II lobbied Congress and with the help of Sen. Barry Goldwater got a bill passed that gave WASPs veteran status.
Mark Gomez, Mercury News
The role of the WASP was as important as the ATA, and enhanced the US manufacturing capability by ensuring that aircraft could be delivered to where they were needed. Without the WASP, aircraft would have either been stacked up at their factories waiting for a ferry pilot or front line squadrons would have lacked combat pilots.
In 2009, President Obama and the United States Congress recognised the importance of their service by awarding the WASP the Congressional Gold Medal saying:
The Women Airforce Service Pilots courageously answered their country's call in a time of need while blazing a trail for the brave women who have given and continue to give so much in service to this nation since. Every American should be grateful for their service, and I am honored to sign this bill to finally give them some of the hard-earned recognition they deserve.
President Barack Obama
Other Similar Formations
Alongside the WASP, ATA and WAAF Canada and Australia both had similar formations comprised of Women.
The Royal Canadian Air Force Women's Division (Originally the Canadian Women's Auxiliary Air Force, CWAAF) was modelled on the UK's WAAF and was created to fill a gap left by men being required for frontline duties. Unlike in the above formations, the women of the RCAFWD did not fly aircraft but instead relieved men of duties such as administrative and clerical work, dental assistants, equipment assistants, weather observers, telephone operators, photographers, wireless operators, fabric workers, tailors, laundry staff, cook. (Source) Their duties were expanded as the war progressed, but not to include flying duties. Over 17,400 women served worldwide, with thirty women losing their lives.
The Women's Auxiliary Australian Air Force (Wikipedia, Australian War Memorial) was founded in 1941 after lobbying by Women who wanted to serve their country and by the Chief of the Air Staff who wanted to free up men for front line service. Some 27000 women served in communications and maintenance roles across Australia. Like the RCAFWD, the women of the WAAAF did not conduct flying duties.
Alongside the mini-conclusions I wrote earlier, I'd like to add that I am almost certain that without the service and sacrifice of the women in the above formations the air war in both theaters would have progressed very differently, without these women the infrastructure required to support and continue the war would have been stretched thin. The role of the combat pilots do, often rightly, eclipse the incredible actions of the women who served in the formations.
Not only did they take on ground roles that men would otherwise have been required for but they ensured that the front line squadrons were supplied with functioning aircraft by ferrying them across countries and by testing them following repair. Also, I feel it would be remiss of me not to mention just how much progress these women made politically and socially within their respective countries.
I can't really find all that much about women serving in the Luftwaffe. However, I have found this website that says that women were conscripted into the German war machine, partially as Luftwaffenhelferinnen ("Luftwaffenhelferin" literally means "air force helper woman", probably "air force female auxiliary" in current US wording. - Jwenting) to provide clerical services before being reassigned to combat/air defence duties as searchlight and AA crews. While they were not technically members of the Armed Forces, they were subject to military discipline and regulations.
The Wikipedia on Hanna Reitsch suggests that at least a few of these women did fly in test and other roles (although, I am unsure about combat roles).
Note: Thanks go to jwenting for reminding me of the WAAF, for pointing out WASP and for translation.