I flew with BA when DD1 was five weeks old. They refused to let me board first, although a bus went off to the aircraft containing only two children who were travelling as unaccompanied minors. (DD1 was the only baby/small child on flight, bizarrely.) I was flying Club Class and had a seat booked and paid for, for DD1. I plonked her in her seat (in travel seat) while I sorted out hand baggage. The Cabin Crew woman said to me "You can't leave her on that seat." I misunderstood, and said "No, that's fine, I understand I have to have her on my lap for take off. I'm just leaving her there while I get organised." She then hissed in my face "No, you can't come on and expect to use a club class seat for a baby. That's not on. There are people in economy who we can upgrade if seats are spare."
I said that I had paid for two seats and showed her my boarding cards, but she was having none of it, and started having a go at me. I lost it at that point and asked her if she was going to make the man opposite move his coat, as he clearly hadn't paid for a seat for it. The pilot had to come and intervene, looked at the boarding passes and said to the woman "What is your problem? She's paid for two seats." I didn't see her for the entire flight, so not entirely sure if she got relocated to economy or simply taken off the flight. The pilot was very apologetic.
I haven't flown with BA since (and that was 14 years ago). I also had a difficulty with a member of cabin crew when I was pregnant, as she was trying to offload me from the plane on the grounds that I was "more pregnant than I was letting on", despite having been shown a doctor's note, and my records, stating I was 28 weeks and fit to fly. The pilot had to intervene then, as well. He laughed at the woman, asked her if she knew better than a doctor how pregnant I was and asked me if I promised not to give birth on his flight. I promised, and one of the other cabin crew (who was lovely) told me that it didn't matter if I did, as she was a trained midwife and would sort me out.