The cooling system of your Series III XJ6 does not pose any particular problem in terms of routine servicing. However, it is essential that you maintain it properly as the XK type engine does not like to be overheated ! Contrary to myth, these cars are not inordinately prone to overheating but are certainly susceptable to ordinary cooling system failures.
The cooling system should be flushed every two years, some owners prefer a more frequent interval. A 50/50 anti-freeze-water mixture is fine although, if you can calculate it this closely, the manual actually calls for a 55/45 mix. The system holds 19 quarts.
The thermostat does not necessarily require routine replacement although some owners do so every few years. If you do replace the thermostat make sure to use a Jaguar replacement or an exact equivalent. Do not use a generic or universal replacement as it won't have the proper bleed holes.
The cooling system has a whole slew of hoses, some of them hidden under the intake manifold. these should be replaced every 5 years or at the first sign of softness, splitting, bulging, or cracking. Remember, hoses deteriorate from the inside out so if a weak spot is visable externally you can bet the hose is on it's last legs. It is also a reasonable assumption that if one fails, the others can't be far behind.
Sometime around 1982 Jaguar went from a metal cooling fan to a plastic one. If you have a plastic fan is very important that you check it often for cracking. These fans have been known to disintegrate catastrophic results. This would be most likely to occur if the fan clutch has seized, allowing excessive fan rpm. Do not gamble here ! If your fan is cracking, replace it and the clutch as well. These parts are expensive, check the archives for alternative replacements if you wish. You can't go wrong with genuine parts but if your budget won't allow it, better to use an alternative than leave a questionable fan on the car. Many XJ6 owners prefer electric fans and there's plenty of discussion on this topic in the archives.
Regularly check the condition of the pressure cap(s). If there is any deterioration of the rubber seal, replace the cap. Caution: Never remove the cap on a hot engine! There is a very real danger of being scalded!
Hose clamps like to loosen up over the years so make it an annual routine to go 'round and snug them up.
Check the coolant level at least once a month (on a cold engine) and, if needed, top it off to the base of the filler neck of the fender mounted header tank. Interestingly, a number of owners report that when filled to this level a quart or so always disappears, presumably into the hidden expansion tank (inside the fenderwell). Leaving the level a quart or so low eliminates the problem and has no ill effect.
That's about it. No magic involved! Also refer to "Overheating Checklist" for more information.