Cadbury in the US is made by Hershey. It has been since 1988, under license, with Hershey's own recipe. The first ingredient in a British Dairy Milk is milk. The first ingredient in the American one is sugar. The American bar also has PGPR and soy lecithin in it, emulsifiers that stretch shelf life.
In 2015 Hershey sued the largest importer of British candy in the country. They settled, and the settlement cut off British made Cadbury, along with UK KitKats, Yorkies, Toffee Crisps and Maltesers. The wrapper looks the same, but the bar is certainly not, and since 2015 the real one is hard to find here.