There were so few LHD export Mk2 cars to destinations outside of North America (USA and Canada) that I would think that optional equipment for a solely Euro destination wouldn't have been cost effective or even considered. Of course, Rootes would have had to made lighting considerations for France and the Continent. France would have required yellow headlamp lenses and Europe would have had to have asymmetrical projecting lenses... Italy would have required clear lenses, in front anyway for indicators, but I don't think any MK2's went to Italy! German cars (and French ones) usually had a supplemental chassis tag and a lot of them also had secondary stampings of the chassis number into the engine bulkhead. Germany required axle weight load information on chassis tags of imported cars after about 1963... French cars often had supplemental noise emission devices for ignition systems (extra shielding and noise capacitors)... Some British cars were supplied with smaller displacement engines (usually de-stroked) to avoid taxation problems for large displacement engines (over 2.8 liters). Of course, this would not have been possible with the Tiger!
There were a couple of cars that went to Rhodesia but they weren't Mk2's... They had Alpine side emblems like the German cars due to the (German) Messerschmidt trademark on the Tiger name.
Outside of the Hartwell 6 there weren't any UK spec Mk2's. My Mk2 was shipped to Ontario, Canada new... But in all respects it is a standard N/A LHD spec Tiger.
I don't think the fan would have been to heat the water up... BMC used to supply fans with more blades to cars destined for hot climates. They called them "tropical fans". it's very interesting that the Healeys from '64-'67 had 12 and 16 blade fans, as opposed to 4, for Germany and Switzerland. Not exactly what I would call hot climates!!
The '64 Aston Martin DB5 I have in for resto right now has a secondary electric fan, but that is because it was fit with factory A/C... It was the '64 New York show car, so obviously N/A spec.