It's been ages since I watched it, but I remember in "Battle of Britain" an argument by two boys about whether you needed to be an officer to fly Spitfires. I don't remember the result of their tiff, but it is less important who was right, really, than the fact it was an argument that might have been had, as it seems to imply that there were important divisions based on rank beyond simply organizational hierarchy. And this being Britain, while certainly you no longer bought commissions, I feel it is hard to separate perception of rank from perception of class, so I don't think I'm overstepping in assuming that the latter was a driving force as much as the latter.
So my question breaks into three broad sub-questions:
How did class influence what and where a member of the RAF was assigned, especially within broadly similar roles: i.e. did you have a better chance of flying a Spit than a Hurricane if you had a hoity-toity accent; and did you have a better chance flying a fighter than a bomber likewise?
Presuming that there was an effect there, how did it change over the course of the war, as the RAF had to expand for its wartime role, and also replace personnel who were killed or invalidated out?
And finally, how much of this was driven by social dictates and expectations, and how much of it was actually built into how the RAF was organized and run?
To answer part of your question I can quote from "The Right of the Line: The Royal Air Force in the European War 1939-1945" by John Terraine.
In that book he mentions the formation of the Royal Air Force Volunteer Reserve the members of which formed a significant part of the personnel available to the RAF in the early days of the Second World War.
The signficant quote is
The Air Council, it was stated in the letter submitting the scheme to the Treasury, "propose to open the new force to the whole middle class in the widest sense of the term, namely, the complete range of the output of the public and secondary schools". "In a force so recruited, " the letter went on, "it would be inappropriate to grade the members on entry as officers or airmen according to their social class; entry will accordingly be on a common footing, as airman pilot or observer, and promotions to commissioned rank will be made at a later stage in accordance with the abilities for leadership actually displayed.
So I hope that throws some light on the first two parts of your question.
There is one other thing I'd like to add which is obliquely relevant.
A very large proportion of RAF personnel in WW2 came from Commonwealth countries and from countries outside of the Commonwealth some of which had been invaded by Germany. Approximately 1/6th of all RAF pilots in the Battle of Britain came from outside the UK for instance. I mention this because whatever class distinctions might, or might not, have been seen between aircrew from the UK the situation would have been different again for pilots from outside the UK.