I did something similar to David L above, three years ago. As an airline employee, I was able to obtain what's known as an ID75 on BA001 LHR-JFK. An ID75 is a standby ticket valued at 75% off the normal economy fare. Of course there is no such thing as economy on Concorde!
Being a standby passenger, I had no reservation, and no privilege over paying passengers. The day I wanted to travel, I was not able to get on the first two flights (they had one that departed at 10:30 AM and a second that departed at 5:30 [I think] PM). The day was so hot, they were WAT limited and could not take a full load of passengers. I tried again the next day after an expensive overnight at the LHR Hilton, and was successful. I sat at 8A, and thoroughly documented the entire flight.
As stated by all above, the flight was spectacular. I won't bore you all with the details here, but if you would like to hear all about everything about the flight, please e-mail me separately.
For those who may not get the chance to fly on Concorde, there is an EXCELLENT book called "The Concorde Story" by Christopher Orlebar, published by Osprey Aerospace. Its ISBN is 185532 6671. You will read about the development, flight testing, and day to day operation of one of the 20th centuries true masterpieces.
Thanks for the memories!