Do tell BA that you have already booked with Swiss. Give dates that you have booked and even flight numbers - to help convince them that they have already lost business.
I don't think they could give a toss. When I withdrew our custom from them, and wrote explaining the reasons why, and included a list of the flights between Europe and SA (about 8 return trips for 4 people, so 32 segments, nearly half in Premium Economy and a couple in Club) which I'd booked on SAA, Virgin, and Swiss, they just replied with a standard 'cut and paste' bull**** letter. My letter was addressed to WW's predecessor Rod Eddington, at Waterside.
They are past caring.