When I worked in BA customer relations, it took up to 4 weeks to get an acknowledgement and then an average of 4-6 weeks for it to be actioned. Who handles your complaint and the timelines involved generally depends on the type of ticket purchased, your Exec Card status, class of travel, etc. A non Exec Card, lead in fare ticket would see the complaint handled in India. A Silver Card holder on a flexible club world ticket is more likely have the complaint first seen in Newcastle. My advice would be to write directly to Willie Walsh. He won't see the letter but it goes to a specialist team at Waterside that are better able to resolve cases like this one. If you're a Gold Card member, there is a direct line you can call for an Executive Assistant which speeds up the process dramatically.
It has been a while since you've been in BA CR then.
Everything sent to Waterside and Gold Card, Premier, Gold Guest List all goes to Newcastle. Everything sent to Willie, Keith or Frank goes to Newcastle. There is no specialist team. The mail is now forwarded straight to the North East
Turnaround for non status is around 10-14 days. Gold's will get a response within 24-72 hours.
Are you bookings 2 seperate bookings? If so, I'd imagine BA will hold steadfast
I think your being unreasonable in your expectation of a reply so quickly tbh