Call some one and get it rewired. It isn't hard but isn't home depot level either. If you were in Savannah I could send you to a guy, and high end lighting store should know someone

this might work its an adapter . otherwise yes new socket.https://a.co/d/9FC9DBW

I agree and BTW your house insurance will not be happy with no UL rating on the fixture so make sure it is done correctly and by some licensed .If it strats a fire you have no insurance.

fissfiss, voltage is nothing to do with a plug.

I've never done a Euro conversion, but I imagine if it's the same 2 wire you can do it (you can even replace the entire cord). You may have to find a socket that has the same neck attachment, or modify the neck attachment, or reuse the same socket shell, but no reason this can't be done. You don't need an electrician. If you have no DIY wherewithal, take it to a mom-n-pop lighting store.

RL, I know…but an adaptor just allows you to plug into the socket…a converter changes the voltage running through the line. You will fry your US lamp if you just use an adapter in the UK.
And if you bring a UK appliance to the US, it won’t work with just an adaptor because it will not be getting enough volts through the line….unless you take an adaptor and plug it into your stove outlet, which is special and the correct voltage. Most modern portable electronics have a dual voltage feature that you are not aware of, so you only need an adaptor.

Re-wire it and breathe easy!

"I agree and BTW your house insurance will not be happy with no UL rating on the fixture so make sure it is done correctly and by some licensed .If it strats a fire you have no insurance."

Not even in the greatest of all nanny states do they issue licenses to rewire lamps, much less require them. And, no, your homeowners insurance is not invalidated by a miswired lamp, whether done by you or someone else. That's just nonsense.

Every American house almost certainlly has dozens of non-UL inspected plug-in electrical devices. (not talking house wiring here, but things that plug into the wall) Read your policy, if it says you're not covered for using non-UL devices (I do read my policies and have never run into anything like this) then I'd definitely switch companies.