I think the article is incorrect, they say that only British Airways and Air France operated the Concorde, but I believe Singapore Airlines did as well.
BA's G-BOAD was painted on one side in Singapore Airlines livery, and jointly operated for three round trips to Singapore via Bahrain; I don't know if any Singapore Airlines crew were involved in the operation.
A more obvious example of another airline is the Dallas to Washington flights (and on to London and Paris), which were operated by Braniff airlines crews within the US (as they couldn't be operated by any non-US airline), and the aircraft changed to US registrations for each of the domestic legs.
The Concord wasn’t allowed to fly at super-sonic speeds while over the continental US, so the poor Braniff pilots didn’t get to really have the full experience (other than in training).
https://airwaysmag.com/uncategorized/concorde-airlines-brani... is an article I found that matches my recolection (at least) for how that worked.