There are some nuances to it. British Airways (not the original British Airways Ltd [0], who merged to form BOAC) was established to manage several existing airlines that had already been nationalised (BOAC, BEA) and two regional carriers (Cambrian Airways and Northeast Airlines).

Of course BOAC and BEA had been made my consolidations of many smaller airlines which gets messy quickly when tracing the lineage. Even Cambrian and Northeast had formed British Air Services prior to this which was 70% owned by BEA.

So it is technically true that is was started as a state owned airline, but one made from companies that were originally created as private with a mixed history of state ownership.

[0] https://en.wikipedia.org/wiki/British_Airways_Ltd