Are you based in the UK? What class are you booked in? Are the Emirates flights QF codeshares or EK flight numbers?
The BA and Qantas flights you can credit to any OneWorld partner, so BA or QF or AA.. However, if it is booked in discount economy, the QF flights credited to BA would only earn 0.25 avios per mile.
Emirates is not part of OneWorld, however they are a Qantas partner. So you can credit those flights to either Emirates Skywards or Qantas. If Qantas, you can redeem for any OneWorld partner award flight.
Yes, I'm based in the UK
The Emirates flight has a QF flight number, but the BA flight doesn't
I didn't realise about different types of economy. I've just checked on the 'manage my booking' page and the displayed travel class is (s), which appears to be the 'discount economy' travel class.
I presume it would therefore be best to put them all onto Qantas frequent flyer programme.