A little before the Napoleonic Wars, the official title of the Scots Greys was "The 2nd (Royal North British) Dragoons", as they were recruited in Scotland and rode grey horses, they earned the nickname "the Scots Greys", this became formallised (officially) later as the "Royal Scots Greys". As for the original question of an English regiment wearing a white cockade, the only one I can think of, off hand, was the Manchester Regiment, who fought for the Jacobite cause in the '45, those that hadn't deserted by the last battle were either sliced-and-diced at Culloden (and it's aftermath) or shared the fate of the highland prisoners (IIRC most of the captured officers were executed except the Colonel, who could prove that he was a serving French officer and therefore couldn't be a traitor to George II).

Walrus