I have done it many times. Both changing planes without a visa and ado booking one way ticket to a country that I can’t stay more than 90 days and another one way ticket with a separate airline. Never I had issues. Everytime I just saw my separate ticket to the check in agent and that’s it. I was pretty sure from the beginning that was BA agents’ fault but I am not an expert on that matter. I am glad the OP is with his family and I truly hope he sues BA or at least he gets UK261 for the whole palava. Further more BA really needs to train their staff properly. From the Indian call center to the check in agents we have seen some serious failings recently. My last experience at T5 when 3 agents spent 45 minutes to rebook a BA flight to another one was quite shocking.