Hello, and sorry to hear about your gearbox "failure".
There are 2 ways to approach this, the analytical, methodical way, or the A.S.S. way (replacing the entire gearbox).
The latter approach is not a good idea to me at all, and not just because of cost. These cars are incredibly complex, and believe me, they have the infrastructure, but skill-wise, they are actually worse off than your FNG. They have the knowledge and the tools to do this but they just won't be able to do it. Worst case the gearbox might fail AGAIN and yet they won't take any responsibility for the same (ask them for a written guarantee on the replaced gearbox for even 3 months, and see their response.) There are many threads here on tBHP about people having their transmissions and engines completely swapped yet they still failed because the root cause was not fixed.
The first option is how I would go about it. First of all take the car back from the ASS because it's a liability keeping it there. Don't crank or run it. Find a garage that is willing to host the car for 2-3 months (discuss and negotiate that beforehand). Whenever you move the car, ensure that the driven wheels stay stationary (use a dolly). Keeping the gearbox in N does not mean you can just push or tow the car with all 4 wheels on the ground (the manual should have instructions). Then perform step by step diagnosis.
First off, I would check the actual condition of the oil + water that has mixed. The oil cooler is a part of the engine radiator, just that they have separate circuits for trans oil and engine coolant to flow. If there is indeed mixing, then it should be there on the engine coolant side as well. Drain the coolant and check it. Take out the radiator, pressurize it with air, and check for leaks to understand how it happened. (You can check what I've done with my Sonata AC's cooling coil
here (My Pre-Owned 2005 Hyundai Sonata 2.7L V6 | Ownership Review, Restoration & TLC).
Let's assume there's a flaw and the radiator/cooler has failed. The next step is to remove the oil pan of the transmission and take a look. If you're lucky, the pan is accessible. Doing this will not mess up any electronics or other hardware and is completely reversible. Open it up and see the extent of the damage. Is there shrapnel in the oil pan? Preserve it and post pictures. Is there anything obviously, majorly damaged in the trans valve body or the oil circuit? If your mechanic is feeling bold, remove 2-3 covers and parts of the mechanism (but ONLY to the point where he's confident of being able to put it back together - nothing drastic!). My hunch is that nothing major should have failed, since there were ZERO symptoms from your side. (The fuse might have saved you actually).
As you may have guessed by now, anything the ASS will do will hamper your analysis (they're sure to have drained the oil already so you won't be able to study that), so it's critical to move the car from there ASAP.
IF there are no major observations to report, it's time to experiment.
Source a good condition radiator first (have it tested if you found a used one). You may not be able to find one specific to the Jag - don't worry, you can use any oil cooler, just that you may have to source an oil cooler for the trans and a radiator for the engine coolant separately, and find some way to accommodate them on the car (just for the testing phase only). Use good quality hoses and clamps, and you should be fine.
Fill 'er up with oil, refill the engine coolant, replace the blown fuse, and start the car (you shouldn't have started the car at any time up till this point.) Check for error codes, listen for any erratic sounds (switch off the car immediately if heard), and go for a drive if everything seems normal. Keep an eye on the engine temp and also the dash for any intermittent error codes.
If the fuse blows again, that's another thing to get to the root cause of. The garage will help you there.
At any point, if things seem off, then you'll looking at a transmission rebuild or replacement. We can discuss that when it happens. But if things seem OK, at best your gearbox's life has shortened but you're OK to go for now.
As an enthusiast who's rebuilt a car from ground up, I know that things are not always what they seem. Given your complete absence of symptoms, I have a strong hunch that your gearbox is at least driveable and that there's nothing wrong in the first place! You'll need to spend a few hours at a garage and oversee this personally (don't take anyone's word for it - you should be there and see the oil cooler and trans oil pan opened up in front of you). But on the flip side, you might end up saving a lot of money.
Good luck.