On oxygen, they will stay as conscious as long as the oxygen flowed. However, at about that altitude, there isn't enough partial pressure of oxygen for the blood to absorb enough. Modern regulators start to feed in higher pressure oxygen to maintain the level. I'm pretty sure they wouldn't have had pressure breathing then - the masks wouldn't support it. They said they had no ill effects, but the msymptoms of anoxia (lack of oxygen) are very innocuous and modern pilots are tought that they may not recognise it in themselves.
If the oxygen failed - time to unconsciousness depends on the individual - but typically at that height it will be in the order of 20 to 30 seconds.
I suspect they would be suffering some form of decompression sickness (DCS) - which would account for the headaches. Modern RAF regs prohit unpressurised flight above 25,000'. And even with this limit - I've experienced the onset of DCS at this regieme. We know a lot more about aviation physiology now.
Of course, they could have been flying Mk VI or Mark VII Spitfires which were pressurised - but only small numbers of these marks were produced.