Is it deliberate?
It is interesting that many large companies do respond to complaints within 24 hours - including some very large companies that have many more customers than BA, and presumably many more complaints.
Now, the companies like BA that typically take weeks to respond nevertheless must keep up with the flow - if they receive 5,000 complaints a day, they must answer 5,000 complaints a day. Otherwise, the backlog would grow constantly larger. But if they can deal with the complaints at the same rate that they arrive, why have a backlog at all? If you can answer 5,000 complaints a day, why not answer the 5,000 that actually arrived that day? You may need to invest in some extra effort to clear an initial backlog, but once you've done that, you can be responding to your customers within 24 hours for the same cost that you previously incurred to respond to them in four weeks.
I can only conclude, therefore, that the delay in BA's case (and in many other airlines, phone and electricity companies, etc) is deliberate, intended to make a certain proportion of people forget about their complaints and go away. Can anyone confirm this, and if indeed it is a commercially sensible approach?
(By the way, just so I can get come perspective on those 5,000 daily complaints to BA, approximately how many passengers does BA carry each day?)