The other alliances don't actually have an 'official' policy in interlining bags. But interlining bags is actually up to the individual airline of whether they will do it.

But Oneworld decided to initiate this and BA is now very strict with this and will not even interline their own flights if they are on separate tickets. CX will tnly interline with CX/KA on separate tickets and I guess they didn't tell you that they too will not interline with anyone else (whether Oneworld or not) if its on a separate ticket.

So ya, you'd have to go through immigration, pick up your bags and then check in with CX. And you better check your baggage allowance with BA inter-europe. Its usually only one 23kg bag in economy.