To EZE, try Iberia as they have many flights to EZE; one of the 2 flights is an A340-300 (the other a 747 I think). From the UK, you will of course fly via Madrid. IB has very competitive fares on this route......if your budget permits, look into a non-published business class fare that is available through travel agents and you may be suprised how affordable it can be. Also, KLM has better fares out of the UK than it does out of AMS, look at them, but that does not give you an Airbus or 773 flight. Finally, Varig flies out of LHR to Sao Paolo and Rio, with connections on to Buenos Aires, again, no 773 or Airbus long-haul, but Star alliance miles can be valuable and it may be your last long-haul trip on the MD11.
Qantas has good fares out of LHR to Australia, again with some attractive business class rates if its in your budget; QF also has packages that allow certain inland flights "for free" depending on your itinerary (a chance to fly on the new QF 738 is possible).......not a creative alternative, but an effecient one. SQ flies direct to Cairns via SIN, not sure of the equipment these days but the LHR-SIN leg will be a 744. You may want to look at carriers flying into Brisbane to get your shot at a 773 and then purchase a seperate ticket up to Cairns.......as Thai, Cathay and Malayasian all have 773s this could be your chance for atleast one segment. Emeriates is also an idea, via Dubai, on to Australia - not direct to Cairns but with a bit of work and a few stopovers it could work out. Lastly, Lauda Air via Vienna has some great fares (nonpublished) out of the UK, via Vienna, to Australia, not a 773 but a 772 instead (does almost count?)
In any case, have a wonderful trip.........I have been to both places and both are wonderful - I am sure you will have fun.