Lee, by all means, all the above things need to be looked into and dealt with. Once that is done, if there is still a notable vibration it could be from the driveline. Since you only notice it between 50 and 60 MPH, I'm inclined to think that may be the case.
Tire/Wheel vibrations are generally noticed to peak at two, and sometimes three different speeds ranges. It sort of rolls in at 30 MPH, then trails off and peaks again at maybe 48-50 and again at 65-70 MPH. Driveline vibrations are at a higher frequency and usually 'roll in ' at one speed range. At 55 MPH, cars with differential ratios like ours, a wheel related vibration will be 12-13 Hz [cycles per second], a propshaft vibration will be maybe 36-40 Hz. Some people can tell the difference, some can't
Jaguar dealers have some equipment they can use to balance the propshaft using special nuts in calibrated weights. I have had some good success balancing XJ and S-Type propshafts if addressing the tires and wheels doesn't yield results. It costs a little money, but it can also make the car a joy to drive if successful.
You might quiz your local dealer to see if they still have the proper equipment, and what the cost would be.
Cheers,