In addition to the factual comments, I better explain how it was for me.
There were 7 staff at Flight Connections T5C, I was their only passenger, so with all of them leaping into action I was through airside in seconds. From airbridge to Starbucks would have been under 5 minutes if I didn't spend so long chatting to the staff. After I left I didn't see any passengers following on behind me either.
The only slight hitch was at the time I landed into T5C my onwards flight was not showing on the monitors, however I was heading on the BA25 to HKG, that's the Airbus 380, and I saw it in place as my inbound flight taxi'ed into T5C. This led to a discussion with the BA staff at the entrance. However they scanned my boarding pass and it cleared "ready to fly" so all was well. This was about 1 hour 40 minutes before departure. If necessary I guess I could have waited for the monitors to update. I got the new text service from ba.com to confirm it was at T5C gate shortly afterwards.
As mentioned above, I did have the option of avoiding this route altogether and going to T5B, in order to get to the lounge, but I know they board the A380s early - or try to - so I decided to head for Starbucks instead.
I was surprised at what a difference it makes having such a stress free transfer who have, it certainly made for a very comfortable start to the trip.