Thanks for the replies, all. I would have replied sooner, but had a very busy day at work.

Soft bags are out, I think, and definitely not just a cardboard box,
@ianrauk. Given that I'll be travelling across the world in September, and the bike is worth about $4500 (or at least that's what it cost me), I'll be using a hard cased bike box, most likely one of
these ***, but I haven't decided yet. If I had any way of contacting BA, I'd ask them to make sure. Perhaps I'll just try their general booking number, then ask to be forwarded to someone who knows what the bicycle baggage policies are - does that sound like a good approach?
*** Yes, I know it's not strictly "hard" case, but it appears to be padded well enough that it makes no real difference.