It's certainly the case that when booking redemptions on non BA airlines, you need to get to ticketing ASAP since some airlines aren't very forgiving in terms of deadlines. Qatar and Cathay spring to mind, JAL, AA and Qantas are more flexible. If booked fully online then automation means there is rarely a problem, it all gets quoted, paid and ticketed within minutes, but if making changes it's going to become a manual process very quickly. When something gets queued off to the back office there are in fact several queues, and there is an express one for non BA redemptions for speedy treatment. That's probably explains some of the comments above, but I would say 24 hours is more realistic than just a few hours. There are a number of problems here, since agents can send a reservation off to the wrong queue, and here we have a complex booking that needs repricing in view of the previous booking. The online tools are limited when repricing over two or more dates so you need a specialist look at the new booking to give a quote, and BA hopes that the reservation will stick until the quote is made, the fare paid and then off to ticketing.

In the OP's case then s/he does need to call every day or two to see if a quote has been made and to push towards ticketing, QR will release availability at some point. Seeing the reservation in MMB is one step, next is payment, then it's ticketing, and you need the last step to be sure you have the reservation if it's non BA. If BA then usually something can be sorted out so long as the traveller takes action before OLCI. Hopefully most FTers know to check ticket numbers if OLCI fails. Mostly it works, but occasionally things fall into a hole and then if it's not BA it can get tricky. One factor here is that the OP is trying to get the back office to respond when there have been several days, nearly a week, of poor weather in LHR and so these passengers are prioritised.

I think there is a general point that making changes to redemptions on some airlines really needs to be discouraged, perhaps the change fees should be bumped up. Sometimes there is scope to just cancel and rebook if there is plenty of availability, but it really is best to get it right first time and online. If people think "oh I can just swap it to this date" then it's often not that simple.