The general timeline seems to be 17-21 days from the date when your old passport is logged as received to getting the approval. Then maybe a couple of extra days for printing/delivery depending on day of week and time of day of the approval. You can phone, plead, beg, cry etc. but although there do seem to be some that get done quicker, unless you have compassionate grounds they do pretty much seem to be processed in date order. And the counting starts from the “documents received” date, not the date the application was submitted. So for anyone sending in an old passport and you are pressed for time, pay for whatever the fastest guaranteed delivery option is. Note that there is also a gap of maybe 3-5 days from the date your passport was signed for and the date it actually gets scanned onto the system and linked to your application. I did Royal Mail 1st Class signed for which took 4 days ( for 1st class!). Then with the extra time to get it scanned into the system, and allowing for bank holiday, there ended up being an 11 day gap between my application date and my documents received date, which was incredibly frustrating.
If your docs were received 27 August then there’s a good chance you have already now been approved.
I had my approval today, 17 Sep, for docs received 31 Aug, processed at Durham.