The 2000 S-Type (and Lincoln LS) used a 5-speed Ford automatic transmission that was considered very unrefined for that class of vehicle. (Frankly, it was considered a P.O.S.) It had a number of reliability problems common to Ford transmissions, prematurely worn bands & clutches, etc.
The car originally came with a 4 yr / 50k warranty, so you have a good case. (The time factor does not apply to the cause of failure.) However, since it's out of warranty Jaguar will only assist if you are the first owner of the car and can prove you've done all the required maintenance services (hopefully at a franchised dealer), not abused the vehicle, etc. If you had the trans repair or replacement done by an independent shop your chances of getting Jaguar to pick up the tab are nil.