Twice!
When the service was reintroduced after the crash, BA offered one-way upgrades for business class tickets to either Concorde or First. I was flying London/NY about every 6 weeks at that time, so, yeah, Concorde, lets do it!
First time was uneventful. Left LHR at 9am, landed JFK at 9am, in the office by 10am. "Oh did you fly in last night?", nah, Concorde... Like you do...
Second was much more interesting ... Having done a bit of homework I now knew that take-off speed was approaching 200mph. So we were barreling down the runway but it didn't feel like no 200 mph. Then the brakes came on, hard, very very hard. Like face on seat in front of you hard. Momentary feeling of panic, what's happening, are we all going to die? Once we'd stopped the pilot came on the intercom in his best Biggles accent and apologised as there had been a technical issue. Apparently one of the reheats hadn't lit up so we weren't going to make it hence the emergency stop.
We then taxied for a while, a really long while. Again Biggles explained, the technicians want us to go to the edge of the air field and try running the engines up to full power to see if it does it again. What???!! This is how you problem solve? We parked up and sure enough the dodgy engine was run up, presumably with the pilot standing on the brake pedal with all his weight. The whole airframe shook and jiggled as it tried very hard to pirouette round in circles. After a few seconds it stopped. "Its definitely broken", said Biggles, "But, good news! We always have a spare Concorde ready to go". Now, that inspires confidence...
We were then de-planed back at the terminal where the BA ground crew clustered around all the proper paying passengers fawning and apologising profusely. I was pretty much ignored since I was on a freebie. True to their word they had us on their spare fairly quickly and we arrived in NY around about the same time that a conventional flight would have. The weird steep approach angle really freaked me out on landing. I wasn't quite so elated by that experience.
I went First next time. Still have the boarding pass for seat 1A.
Yes, the exterior walls did get notably warm to the touch. The toilets were very warm too. Wish it had been during the era of smart phones because I have nothing to remember the experience by except my own unreliable memory.
Also remember being parked on a taxi way at LHR waiting to take-off with a view down the runway when a Concorde took off. The jet wash really rattled the side of our aircraft as Concorde disappeared in a yellow haze of its own fumes. That's the other reason it couldn't be in service today.
I've seen 3 of the stuffed and mounted ones, been inside two, wonder if one of them was one of "mine"..
Ah, the days when we actually could claim to be World Leaders (at some things).