After pondering the title of this thread for a bit, I can't help but to wonder if the concept of antipodes even applies in the context of a flat earth map. What exactly does it mean to have opposite points on a flat earth?
Well, quite. It would be a pretty meaningless concept. But the fact that you can get flights from London to New Zealand via multiple routes is pretty much the point. On a flat earth there is only one shortest line between points A and B. On a globe if the points are antipodal then there are multiple shortest lines and you can put those lines through any point C on the globe.
When I've got more time I'll have a look at flight times between antipodal points via different "C" points and do some comparisons.
In theory they should be reasonably consistent and that wouldn't make any sense on a flat earth.
Just to kick things off then, looking at bookable options, I picked a few extreme examples, all on the same day, starting from London Heathrow (LHR), one via Los Angeles (LAX), the other via Tokyo (NRT). The ultimate destination in both cases is Auckland NZ (AKL).
LHR and AKL are not ideal because the antipode of LHR is actually about 1600km (1000mi) S of AKL. There are other pairs of locations in the UK and New Zealand which are a better match, but we need major international airports with lots of flight options and these two are the main ones. What this means is there will naturally be some variation in distance (and hence flight time) depending on route and prevailing winds, but when we're talking of flights approaching 24h and 12500 miles, the differences aren't that great as we shall see.
Having done a bit more research, I think Madrid (MAD) in Spain and Auckland (AKL) are actually a closer match as the antipode of Madrid is only 400km (245mi) from Auckland and Madrid is an international airport, so although there are fewer flight options compared to London Heathrow, I've also included a couple of flights between the two, one via Beijing (PEK) and the other via LAX.
I looked them all up on flightradar24 to find actual route and flight times. Distances given are great circle distances from the flightradar24 pages:
A) 31st Oct BA269 LHR -> LAX flight time 10h 4m, distance: 8780km, followed by AA83 LAX -> AKL flight time 12h 11m distance 10467km (total flight time 22h 15m, total distance 19247km)
B) 31st Oct BA5 LHR-> NRT flight time 10h 46m, distance 9615km, followed by NZ90 NRT -> AKL flight time 10h 2m, distance 8806km (total flight time 20h 48m, total distance 18421km)
C) 31st Oct DY7743 MAD->LAX flight time 11h 25m, distance 9406km, followed by NZ1 LAX->AKL flight time 12h 1m, distance 10467km (total flight time 23h 26m, total distance 19873km)
D) 31st Oct CA908 MAD->PEK flight time 10h 30m, distance 9266km, followed by CA783 PEK->AKL flight time 11h 58m, distance 10402km (total flight time 22h 38m total distance 19668km)
The London flight via LAX is longer than the flight via Tokyo. The difference between the two is 1h 27m in the air and 826km (513mi). This is a relatively small discrepancy, explained by the fact that they are not exactly antipodal, but they are close enough for the two routes to make economic sense.
As expected, the two Madrid options are closer still, 48mins and 5km.
On different days with different winds and routing, timings and distances will vary somewhat, but not substantially. Bear in mind that these are also extreme examples of routes spread a long way apart.