I am afraid the answer is probably a bit of everything :)
At home in Blighty - the 'proper' RAF Stations would have a Station Laundry but I doubt that the smaller and/or more temporary Stations/Airfields did.
If no Laundry available then it would be up to the individual to either DIY it or arrange for (say) some local ladies to do it for a small fee.
Abroad of course in certain countries many pilots would have some form of personal or shared servant who would either do the laundry or arrange for somebody to do it (ie 'Dhobie Wallah').
A letter home by Flt Sgt Roy Fennell 8/2/1944
Sorry about the bold black image above - the page I clipped it from was black with white text :(