I did read that, Terry, and it was part of our horological history at school. We were told that many companies laid claim to the first wristwatch 'invention'. Breguet himself reported a private sale to a wealthy woman in 1821, I believe.
Personally, despite the claims, I find it hard to believe that a jeweller or watchmaker would produce a wristlet style watch as far back as 1571. Watch movts at the time were massive things, and the watches were called 'Onions' and worn at the belt. They were treated as oddities and fanciful pointless gadgets, even by Samuel Pepys, almost a hundred years later.
The description of the armlet with a 'clocke' within is in reference to a gift from Robert Dudley, Earl of Leicester, in 1571. He rose to the peerage in 1564. Descriptions of the gift include one that describes the 'clocke' as hanging from the armlet.
Since it no longer exists, it is difficult to say exactly what this was. It may have been entirely decorative.