The transmission in the S-Type is a variation of the Detroit Gear 250 that was used in Studebakers an some big Fords (50's models). It is a reliable transmission for the most part but very heavy and inefficient. If reverse is inoperative ,first check the linkage. I have found that unless the selector shaft is fully in the reverse position that it will not engage. Also it takes more fluid to make reverse happen so overfill if necessary to check it. Checking the fluid level in a DG250 is a different process than most other automatics.
Parts and a knowledgeable rebuilder may be a difficult find. I had to ship one out of a rare XK140 FHC out of state and it did not function correctly when installed and had to go back to the rebuilder.
The floorpans are different from auto to manual. I did convert a 1965 3.8S from auto to manual about ten years ago. The rear mount is different but not a big problem to modify. The biggest difference is the console which is considerably wider than the manual car so I had an upholstery shop modify the auto console for a shifter and it came out well. The pedal box can easily be converted for a master cylinder.
Now for my opinion. If it was my car and it has to have a new transmission then I would find a way to install a GM 700R4 in the place of the DG250. I installed countless TH350's in XJ6's in the eighties and nineties but the absolute best swap was the 700R4 in a Jaguar MKIX. I installed the first kit from John's Cars (he admitted that he had not converted one himself) and it totally transformed the personalty of the big sedan. The lock-up converter dropped the engine RPM down at cruising speed. I converted another MKIX shortly thereafter.
A correct short-tail full syncro overdrive gearbox is typically difficult and expensive to obtain. They only came in 65-67 S-Type,420's a handfull of early XJ6's. Late MKII's had a longer overdrive that might fit the S-Type but I don't know for sure.
I've logged in about 60,000 miles in my two S-Types and they are a pleasure to drive if sorted correctly but remember that they have a Jaguar logo which means that they have an appetite for $100.00 bills.