I'll tell you about my car's old/original radiator, but I am not familiar with the XK-150 radiator, however, I don't think Jaguar used different radiator technology in those days, they all used what my radiator shop called a "V" core, which is no longer used today. (obsolete). I have attached a picture of my rebuilt radiator, which I'm sure is very similar to yours. The Core measures 17" x 17".
The radiator shop measured my old radiator, called a Core supplier, (I was there listening), and then told me it would be best to use a Double-Row core, which is the modern way to go.
I paid $300.00 for the Double-Row core and their labor which included pressure-testing the new radiator and painting it black!
I did remove it and re-installed it myself, so I saved a bit on the R & R job, not to mention I did it very carefully because it's my car and I wanted it perfect. I also installed a new Water Pump and Thermostat as they and others recommended. (mine were very old anyway). Before filling the system, I did a Prestone Super Flush, drained the system, then did a water flush, drained it again, then filled it with a 60/40 mix of Prestone Coolant. (60% Coolant, 40% water).
The result? I have noticed the engine does not get as hot as before, the temperature is more stable and predictable even in bumper-to-bumper traffic, I like to drive my 'S' type no matter what! Perfect temperature? no, the 3.8 engine runs hot no matter what, but it is better now, much better.
I am not putting down aluminum radiators, only the price. My understanding is that the small benefit you get from an aluminum radiator is not worth the extra expense, and with the savings, you could install an Electric Fan kit...