Qantas Airlines covers Australia and New Zealand
While our plan for our trip does seem to change every few days, I feel like we are narrowing down our options. After chatting with multiple travel agents in both the US and London today, it looks like the following makes a lot of sense:
London – Hong Kong – Bangkok overland to Singapore – Cairns overland to Melbourne – Christchurch overland to Auckland – Fiji – Honolulu – Los Angeles – London
Will this be our plan tomorrow? Who knows. Sounds good today, though! 
I have no idea why it is so much cheaper to take a multi-stop trip out of London. So much cheaper, in fact, that it more than pays for the trip to London. Not only that, if we buy this package deal (for example this one from Qantas Airlines) we also have the option to add additional stops in Australia and New Zealand for very reasonable prices.
I am so excruciatingly jealous!!!!!!!