I know there is no point rubbing it in now, but in my view, both the time you advised them to get there and even more the time they did was way, way too late if they didn't have their BP and in those circumstances. For all the reasons mentioned above (and 13901 makes an interesting point), your wife and children were not in a straightforward situation.

Again, I personally disagree that airlines can just stick to the "on that ticket" basis, but in the absence of a single ticket, the onus is absolutely on the passenger to be able to prove that they are self-connecting and that the rules corresponding to the individual ticket should not apply. BA do not have to trust you (for the reasons 13901 mentions) and to clarify something, the reason the risk was much higher for the airline here is precisely that despite your description here, your wife and kids did not have a connection. A single itinerary and connection would be recognised by everyone - including immigration authorities - as indicative of the final destination of the passenger. Separate tickets are not and again, in this case, your wife and kids were effectively asking BA to override the final destination as it appeared on their ticket by taking into account another, separate ticket which BA had no knowledge of and had to take the responsibility to take into account. As hinted, if this had been a fake and your wife had tried to enter ZA without a visa and been rejected, the ZA authorities would have squarely blamed BA for it and imposed whatever fine they impose in such case plus the price of flying your wife back. If the ticket had been LHR-JNB-MRU (single ticket) and your wife had tried to make it into ZA during her transit, the ZA authorities would not have blamed BA nor fine them. That is essentially the (big) difference.

So all that to say that as you yourself say, I think you and they underestimated the complexity of the situation and I agree with KARFA if you had asked here before, virtually all of us would have been worried about the prospects of the enterprise and advised for ample caution and plenty of time at a minimum.

And yes, that is something airlines do routinely. If there is a situation in which they have doubts as to whether a passenger will be allowed into a country, they have contacts in immigration departments in arrivals airports and can and do ask in advance whether they should let the passenger proceed or whether they would be denied entry and should be prevented from flying. In this case, the situation is a bit different in the sense that it was not about asking whether a passenger would be allowed through immigration but whether they should be allowed to fly based on their intention to self-connect and transit rather than exiting as per their ticket, but I suspect BA would have had no issue flying and ZA authorities no issue answer. You need to put yourself in BA's shoes here, they are not trying to make your wife's life miserable here, just to avoid being in a situation where they are blamed and penalised for allowing people in who should not have been allowed through, so what they need is some due confirmation that allowing the passenger is ok to cover their back. You are sticking to the idea that there is no material difference between a connecting ticket and someone buying a separate onwards ticket are not different, but unfortunately, it is not so, and in that sense, BA as an airline - and the person who could have authorised this despite the system saying no - needed the right paper trail to confirm that they had done due diligence and should not be blamed for letting your wife fly. The Timatic printout was simply not good enough for that, but an ok from ZA authorities if they had been willing to confirm it would have been, and by making that suggestion, your wife would have proposed a route to the BA agents which, in all likelihood, could have resolved their problem.

Finally, I fully take your point on this being late and last minute, but precisely, the last minute nature of the whole thing could not be blamed on BA and could only result in dramatically increased suspicion.

I think that it is important that you look at the situation not from your own point of view as someone who actually knows that your wife and kids only had good intentions but from the point of view of a random BA staff who doesn't know that. If I were that person, I'd have asked myself:
- If those pax want to go to Mauritius, why did they book a flight to JNB and then a separate flight from JNB-MRU instead of a single ticket?
- OK, maybe buying separately was cheaper for example. But if so, why did they book the LHR-JNB flight ages ago and the JNB-MRU at the last minute? On the face of it that largely wouldn't seem to make sense (even though you have explained to us why here)
- And given the complexity, why would the passenger arrive merely 1 hour before departure which is incredibly late? Is it because they want to put me under pressure to have to edit their BPs without having any chance to check their story and having to take it at face value instead?

So the last minute purchase plus last minute arrival at the airport will combine into furiously ringing alarm bells and unfortunately that late arrival also gives BA a perfect excuse that they simply did not have time to check whether conditions were confirmed that should have lead them to allow the passenger to travel and that they recommend people arrive at the airport 3 hours early even without any particularly tricky circumstances so they will quite likely hide behind that late arrival too.

PS: on the idea of going to Court and building a case, if that is your intention I think it would be worth thinking about it quite carefully as my amateur perception is that this is far from a straightfoward case, and regardless, knowing the name or employee numbers of people would have actually zero impact on your chances to win or not so again, from my point of view, that is very much the wrong strategy here even with that additional context. Ultimately, BA was never going to argue nobody told your wife she couldn't fly, they would just argue that they were right to tell her so in the specific circumstances.