.... As mentioned above it is important to have the correctly rated fuse. The majority of devices here including lamps require a 3 amp fuse, and only high current appliances such as kettles, toasters etc come with 13 amp fuses. ....
Ahhhh, yes - this is another difference in UK style plugs, they have a built in fuse, uncommon in US style plugs.
I think the UK approach is better/safer. And it is why (despite some protestations from someone with an MSEE), I mentioned that there is a
chance that the lamp wire may be a small gauge, and
could, under an internal short circumstances, create an unsafe condition where the wire
could get hot and start a fire, or other heat damage before the panel circuit breaker would pop.
An exception in the US, which will make this clear to all, are the little holiday lights we use (either LED or small glass filament bulbs). Those use small wires, to make it easier to decorate (and cheaper). And you will notice,
these do have two tiny fuses in the plug connected to each of the two prongs. That is for the exact reason I mentioned - if there was a short somewhere along the wiring, that small wire could get very hot (similar to the heating coils in a toaster) and create an unsafe condition. The plug fuses help to protect against that. Small wires without a plug fuses can be a hazard.
You could possibly have a similar situation with that lamp. As Alan says, 3 amp fuses are common on UK lamps - so it is likely that they don't bother rating that wiring for much more than 3 amps, and it will be plugged into a 15A circuit here in the US! Safety first, always. No one should be giving this very real concern short-shrift, especially an MSEE who has experience with house wiring. Sorry, but I will not "hold my tongue" when it comes to personal safety.
But replacing everything, and bringing all components up to compatibility with the way things are done in the US, as mentioned several times in this thread by several people, is a safe approach.
-ERD50