Transatlantic movement of people and ideas goes back to the eighteenth century, and to commercial technology and engineering to the nineteenth.
 
Lavoisier, the great French heat physicist, executed during the French Revolution, whose widow married Benjamin Thompson (aka Count Rumford who transformed French and English fireplaces), and who discovered black body heat radiation in the 1780s - did not invent the product or the word 'radiator'.
 
It appears that the word and the product were invented by Denison Olmsted (1791 - 1859), a scientist who produced the first U.S. geological survey, who named the Leonids meteor showers and who, in 1834, designed and built the Patent Olmsted Stove which was sold both sides of the Atlantic and featured two radiators.
 
On the patent on 5th November 1834 Olmsted wrote: For distributing the heat, I use, instead of the open iron pipe generally employed for that purpose, a peculiar kind of apparatus, which I call a radiator, constructed as follows …
So this seems to be a definitive answer to the question who invented the first radiator, which started as a hot air circulator, went on to become steam powered, and finally became a means of transferring heat from fuel to the inside of a house using hot water. And which now, with the advent of seriously well insulated air-tight buildings, may become obsolete in the not too distant future.