Really interesting thread. Though I don't fly BA, the information here is quite nice to have since I do use these services often to reduce waiting to a minimum. Some of the ones that I find useful:

• CDG - I've been using https://swiftvipservices.com for my fast track needs. Their agents always skip the security and immigration lines to go straight to the front. There are other providers like Chabé, and Extime, but their agents don't always reliably skip lines.
• AMS - https://schipholmeetandassist.com offers fast track through passport control, at least during transit. If you want a true VIP experience, with tarmac transfers then that's also possible via https://www.schiphol.nl/en/vip/.
• GIG - The airport offers the service themselves: https://www.riogaleao.com/passageiro.../hospitalidade.
• GRU - This is more akin to a VIP service, and charges accordingly, but if you really want to skip immigration too, then this works really well: https://terminal.btgpactual.com/en.