Stilton -
Pre-accident I think we did use different tyres than AF. I also recall that BA elected to not use retreaded tyres while AF did, but am not 100% on that.
A pivotal part of the return to service was the Michelin 'NZG' tyre. (Near-Zero-Growth).
The tyres on Conc were incredibly hard-worked, partly because of the speed and partly because they took the full weight of the a/c throughout take-off (a conventional wing is producing a fair bit of lift prior to rotate - concorde produces none of note).
A LOT of energy is stored in a heavy tyre rotating this fast, so a burst can shed debris at great velocity.
The make-up of the NZG meant that it contained the expansion caused by rotation better (so less stored energy in the carcass), and had a far more robust and damage-tolerant structure. The videos of the destructive testing compared with the original tyres is frankly amazing.
The tyre was being developed by Michelin for the A380, I believe, and the principle was adopted for new Concorde tyres. In my opinion, this was the contribution which ensured we got back in the air.