So, it seems.
Actually I was mislead by this units nickname into thinking that they were meant to be the !st Dragoons (The Royals). However, I was puzzled by the Colonel's name as the officer commanding the Royals at Waterloo was Lt.Colonel Arthur Clifton, and the officer named in the game is Colonel Fuller.
It turns out that Lt.Colonel William Fuller commanded the 1st Kings Dragoon Guards at Waterloo, not 'The Royals'. So not only is the jacket the wrong colour but the unit description is incorrect too.
Not that it isn't easy to get these two regiments confused. They are both dragoons and both numbered the 1st, except that one is a guard regiment. Not only that they were also wore very similar uniforms, which even caused confusion on the day. Captain Kennedy-Clark of the Royals relates in his letters that during the advance of Ponsonby's and Somersets Brigades the Royals and Kings became intermingled due to the need to circumvent the farm of La Haie Sainte, and in attempting to reform his troop he shouted an order to some nearby dragoons whom he assumed were from his squadron only to be rebuked by them that 'We are Kings, not Royals.'