Richardcb
Welcome to the forum and congrats on your purchase...sounds great

Year/ Colour???
Re original fit tyres, too right, suggest you replace them immediately
I'm pretty sure the Dunlops are no longer available.
You may wish to try
Vriedenstein Quatrac 2 (sponsors of XJS Racing)
Michelin XWX
Pirelli P4000
I used to run on the Pirelli's, as recommended by Jaguar for the XJS. You should be able to get them for a hundred pounds or less all in from your local dealer, tho he may tell you they are not available..........this is not true. Rather they are rarely purchased and so are not stocked in most places and they probably haven't sold any for years.
In fact they ARE available by special order from Pirelli, should be following day delivery..
I can't comment on the Vriedensteins or the Michelins but the Pirellis are good tyres, good grip, no tramlining problems etc. and reasonably hard wearing...I would expect to get 25,000ish out of them.
You may, however experience problems if you mix tyre makes/types, especially on the same axle
Take a look at various tyre discussions in the suspension section.
Best Wishes
DH Dove