Wiring up a light fitting made for the US market to a UK power supply
Start with the UL listing (UL is America's CE, our authority which tests and lists fixtures.)   Is the fixture listed for 240V?  We do have 240V in America, but it's not commonly used for luminaires.  277V is, but not for E26/E27 Edison bases. 

Our trade agreements require we honor each other's listings, but that does not mean the fixture is listed for 240V.

It's possible. Insulation is the easy part, and most insulation in America is 250V if not 600V.  Higher current adds real cost, but since you are doubling voltage, you are halving current, so that's not a factor here. Besides, most of our stuff comes from China, and much of that is also sold in other 240V countries.   

One option is to get a mega-ohmmeter and "meg out" the wires at 400-500V.  It's likely to pass.  That won't make it legal, but it would speak to safety. 

Another option is rewire the lamp with UK-listed or 250V-listed sockets and wires, assuming they are physically-compatible, i.e. the apocryphal '3/8"' threaded rod used to tie most lamps together.  

Also, tungsten bulbs? Really, check out LEDs - they have gotten quite good of late (if you disregard cheap junk, which is still junk.)