Just had a few data points to share on this as we were scrambling and moving flights around before finally getting the right one.

in general, the best thing to do is to ask to speak to someone in privilege club. However, there still is some luck in regards to the actual agent you get.

we had several cancellations we needed and here were the results:

#1 - female agent with privilege club assured me and guaranteed me that we would have the avios accounts within 30 minutes of hanging up. She put me on hold for 3-4 minutes to do the cancellation and process the refund. The points were back in my account in 15 minutes.

#2 - male agent with privilege club originally told me it could take several business days but then I press him and he relented and after putting me on hold for 3-4 minutes I had the avios back in my account while still on the phone.

#3 - after getting the saver rate I had one more cancellation to make. A male agent would not guarantee me any time frame. Since I already had my new flights booked with saver and no longer was in a rush to get the points I didn’t push back too hard. I did mention the other experiences but he seemed a bit less knowledgeable. He processed a refund “request” almost instantly without putting on hold for 3-4. It’s been 24 hours and I still haven’t gotten back the avios. I called later and spoke to a manager who seemed befuddled by my recounting of experience 1 and 2.

seems like there’s no consistency, but I’d recommend trying privilege club until you get someone who knows what they’re doing and is willing to promise you quick refund. My gut says there’s a refund request, which takes a few seconds. And then there’s actually processing a refund which requires leg work and a few minutes hold.