Does anyone know what my best option of travelling half way around the world is? All these companies seem to offer RTW tickets but I'm looking to fly:
Auckland > Buenos Aires
Buenos Aires > Rio
Lima > Mexico City
Mexico City > New York
New York > London
My only option at the minute seems to pay for individual flights which obviously is more costly 
Can any1 recommend any companies that offer alternatives to RTW tickets
It is either a round the world fare or sector fares which cost a lot of cash when South America is included.
Your best bet would be to look at getting a rtw fare using the One World Alliance. With this you would need to fly from Auckland - Santiago rather than Buenos Aires.
After you reach the London part of your itinerary, just make a fake remainder of the itinerary which you have no intention of turning up to. This would allow you to have a valid round the world ticket and still spend a lot less money than if you do sector fares.
I know that when I was planning my next trip to Ushuaia in southern Argentina (amongst other places) it was a few hundred dollars cheaper (before taxes) for me to get a 4 continent rtw fare stopping over in Africa and Europe on the way to South America than it was to buy a simple return from Sydney - Ushuaia flight.
If you chose to get the rtw fare but just not turn up for the last few flights that you put in as fakes to make a valid ticket, you would just need to make the last section of the fake itinerary as direct as possible so that you don't pay more taxes than necessary.
Depends where you want to go, book a flight to Bangkok and from there you can really get anywhere cheaply
aharrold45 has great advice! One World charges per continent and does included South America, but you have to return to your original departure city. If that isn't a problem, One World should be your best ticket. Try flying into major airport hubs... each connecting flight counts, and so does ground travel.