There are two kinds of oneworld tickets, one limited by segments (one world explorer) and one limited by milage (global explorer). I'm guessing the former is better for you so I'll talk about that.

Vietnam is explicitly forbidden on these oneworld tickets for some reason, and I don't think Laos(Vientiane) flights are available on Oneworld. The cost of those flights isn't horrible if you fly from Bangkok, already one of the stops on your ticket.

I came up with an itinerary and added some needed intermediary cities. The // indicates overland surface sector connection.
LHR-MEX-DFW-UIO//LIM-CUZ//ARI-SCL-PUQ-PMC//SCL-AKL//ZQN-CHC-SYD-BNE-SIN-BKK-LHR. That's only 14 flights on a 5 continent oneworld explorer, you can do 6 more. You could squish in all sorts of things like Sri Lanka or Japan, or lots of places in North America. Just make sure that you don't have more than 4 flights within a continent (6 for North America). If you really need extra flights within a particular continent, you can pay a reasonable fee to get up to 2 extra (still 20 maximum flight numbers overall for ticket). If you really want tomaximize the ticket you could have 20 segments after buying the cheap segments seperately, like lima to cuzco, or some of the ones within NZ and within Australia.

I added an extra PUQ-PMC over what you listed because I was skeptical you will really do PUQ-SCL overland once you know what is involved. There is no road between these! There is a boat, very rough seas, a real chunderfest, and not that cheap. I suggest flying from punta arenas to puerto montt (PMC). You could easily overland to santiago from puerto montt, and it's so beautiful there. Or just use one of your 6 remaining flights to fly PMC-SCL.