The issue here is we really don't know how BA behaved. We have Mr. Banner's explanation:

"Eventually they gave us boarding passes but did not indicate that the nanny, whom their mother and I needed to look after the children whilst we worked on the flight, had been downgraded to a separate cabin from us. We were only told this at the gate on boarding the plane".

This version of events is compatible with (a) the nanny was given a club class boarding pass which was substituted at the gate for an economy one, or (b) they were told that as matters stood Club was full, but they'd have the Nanny on standby for a club seat for a no show, and at the gate they found that there were zero no shows.

Version (a) - yes BA probably shabby.
Version (b) - BA had warned them and not so shabby, (at least in the context of warning them).