A rumbling sound has developed at 40 - 50 mph What might it be?

I would ask to go for a drive with the technician working on the car to point out the noise. Tires can make noise (I recently put a set of tires on my own car and am unsatisfied with the amount of noise they make by comparison to the last set of tires I had on the car). If the tires that are on your car presently are being blamed for "tire noise" I would think they would have been making noise all along. If they said you have tire wear did they align the steering and if so did you see the alignment specs before and after alignment? Do the tires exhibit signs of tire wear? I would be inclined to think you have wheel bearing noise.

This noise is common to nearly all Jaguars at some point and only gets worse as time goes on. I have 3 jags at the moment and they ALL do it. This is a drive train problem having to do with the universal joints which connect the differential to the whichbone assembly on each side. These "U" joints are notorious for rumbling at a particular range of speeds, I'd say from 50-66 mph. The rumble seems to go away at higher speeds although the increased ambient noise at the higher speeds could simply mask it at that point. I've rotated, changed tyres, as well as replaced wheel bearings and wheels and it has never affected or changed this sound. Jaguar has a tool that measures low frequency vibrations as the car is in motion, it was designed for this very type of noise detection. I may be full of bull here, but this has been my personal experience after having owned over a dozen Jaguar sedans from 1972 - 2001 models. I have never gone so far as to change the universal joints, I've decided instead, to live with it.