First of all, sorry for your wife and daughters - this must be a frustrating and tiring experience for them, and I do hope that the option of the nonstop MRU today will work out as it sounds like the best resolution right now.

It's one of those cases where I personally think that BA was mistaken but also where the organisation of the trip was always going to lead to significant trips of such difficulty occurring and where, if I may, I'm not sure your own approach to the issue was optimal.

Let's start from the BA worries here - first, I think it was an excellent idea you had to ensure the transit was at least possible. BA would of course potentially worry about what happens if the family tried to get into South Africa (from that point of you, your comparison with any international transit is not entirely apt in the sense that booked as a single ticket, nobody could fine the airline if someone tried to enter a country at transit point, here, because they sold a contract to ZA, they are more exposed and would worry about being potentially fined), and for example if the flight to JNB was delayed in such a way that the family did not make the MRU flight and would be stuck in JNB.

I think that's the source of the issue, and I think that one prudent thing to do in such case is to call BA beforehand to ask that a note be put in the booking about the second itinerary. They will still be separate and not protected in any way, but at least it gives a paper trail that the intention of the family is to travel to MRU, not to JNB, and it might help in discussions at check in.

The second thing that would help in such case is to go to check in very, very early. You mention that the clock was ticking and conformance became an issue - can I ask when your wife and kids went to the airport? In such a case, I'd recommend going there at least 3-4 hours before the flight as the sort of issue they experienced was somehow predictable in my view even though, as mentioned, in my view, BA got it wrong as transit without visa rules don't usually require that the transit be on the same ticket.

The third point is that if it happens again or to someone else, in my view, and without wanting to sound critical, your own focus in the resolution was perhaps not optimal here. I think that the reference to Timatic was right. However, I just think the whole thing about trying to get people's names, employee numbers, etc is completely counter-productive. First of all, it adds exactly zero to your case (if you are going to complain, you absolutely do not need any name or employee number nor do they add anything to the complaint, and your issue here should be with BA not with a specific employee) and they are openly threatening and will never put the people you deal with (or in this case, your wife is dealing with) in a mood when they want to help or find resolution.

I think what might have helped in this case is for instance to suggest that BA get in touch with South African immigration authorities to get confirmed clearance that the plan raised no issue for them. Chances are they would likely have done that and authorities would have undoubtedly confirmed that even on separate tickets they would have had no issue with this itinerary as your wife would not have needed to enter South Africa. But all those things work a hell of a lot better if there is plenty of time and if the atmosphere has not become antagonistic.

At this stage, I think that realistically, you have two options here.

Option 1, and the one I'd pursue is having your wife and kids flight to MRU today on the nonstop and enjoy the holiday together even shortened by 1 day. When it's all over, I'd submit a claim for denied boarding for the three pax under UK261 (and hotel costs if they needed accommodation for the extra night in London). This might work or lead to a fight with BA to recover costs, but at any rate, you'll have already cut the loss and enjoyed the most important part which is the holiday together.

Option 2 if you prefer is to cancel the trip, ask that their tickets be refunded and again seek denied boarding compensation. It might take a fight too and again, I doubt BA would refund you any of the other things you are talking about (you could still travel, you could still use the hotel, etc) but I think that in the end in my view, you are likely to get the cost of those unused tickets back.

Obviously, it's either/or. And obviously as well, BA shouldn't really be held responsible for the frustration of the initial visa debacle so I'd personally stick to the BA related issue (wife and kids were booked to JNB on BA but with an onward separate JNB-MRU which they intended to fly without ever entering ZA. They were denied boarding at LHR because wife did not have a visa for South Africa but that was not needed as she just intended to transit without visa and that is allowable including when transiting on separate tickets) rather than risking confusing matters by mentioning the failed attempt to get a South African visa in the first place etc.