Not clear from your post, but if you mean the lamp currently has 3 core cable, then it probably has acccessible metal parts, which means it requires a safety earth. Replacing with 2 core cable would be dangerous in the extreme.
American voltages being half of ours mean that their conductor has to be twice as thick as ours to carry double the current. Their insulation has only to be half as thick as ours because they only have half the voltage that we do to drive the current along the conductor. So basically, their conductors would cope but their insulation may not. Their lamp socket is probably also 'Edison Screw' which you can get hold of from lotsa places now but more expensive for replacements. Best replace socket and cable to BS.