I have flown to Oz via BA and Qantas. There's not much between them in terms of service quality. I think you get more water being served on the QF flights. I have only flown economy with BA, economy with one sector business class with QF (thanks Feather #3)
I was told that (at the time, early 02) that not all QF 744's have tv screens in the seats. All of BA's do.
If you can, go via Singapore, a much nicer terminal and facilities than Bangkok.
Internal flights in Oz can be expensive, even with Virgin Blue around. So, weigh it up, enquire about multi stage internal flights with QF once in Oz. It might be worth the 100-200 quid extra. (esp if the kids are planning a "Sydney-Perth" trip)
Another thing, try and avoid the mega popular times to fly. You can sometimes get a few spare seats to have a kip on. The 14 hours between LHR and SIN can be boring...
Top tips. Get up and walk as often as you can. Take compression stockings (yeah I know, they look uncool but so does fat ankles). Take batteries for the walkman, ear plugs for sleep and extra water.
I haven't flown Emirates, a mate flew with Royal Brunei from Perth Oz. One extra stop that added quite a few hours to the trip + no grog.