My experience is that there are more Qf flights available for QF miles. I have been lucky in getting seats on QF using either BA or AA miles, however it is luck rather than anything else.

June or July would be peak periods out of the US so I would not hold my breath. Seats ARe released and if you have status with BA, you could see if the BA NA rep can check (hassle) for you.
It DOES happen, but I would stick to the original trip plan if at all possible rather than hope for seats to be released.