BA use CLC with a few hubs around the world (Gatwick, Berlin as well as a few others I can't remember of the top of my head.) Information is sent by telex to the relevant desk at a CLC hub and the final loadsheet is sent via ACARS (or ground to air radio for non ACARS a/c) during the taxi out.
I'm not sure what department it comes under but the majority of the time it works well. I preferred the days when dispatchers did their own load control and loadsheets, however. It was just one less link in the chain where potential delays could occur.
Just to add, I think Servisair have their own CLC service with a hub in MAN for the UK operation. So although even individual airlines might not officially use CLC, all their loadsheets are prepared remotely. However I believe they're still handed to the flight deck before pushback like the old days.