Nassau was not just a place, it was also a noble house, and had marital and dynastic ties to the house of Orange, hence the Principality of Orange-Nassau. In 1815 though Willem IV traded Nassau with Prussia for Luxembourg, and it was then ceded to the independant Duchy of Nassau.
Therefore, the two Nassau battalions at Waterloo were under Wellington's command, but not as part of the Netherlands' armies, they were simply post-Confederation anti-Bonapartist forces - contributing to the Seventh Coalition. Their commander, von Kruse, had actually fought against Wellington's forces before in Spain from his time in the Confederation of the Rhine, but they could apparently put the past behind them for the sake of the present.
I feel the need to correct you here:
There were eight battalions of nassauers in the 100 days, plus a volunteer company, not just two.
The
first and second regiment
both had the orange cockade, but I'm not entirely sure what their status was.
The '3rd' regiment of Nassuers, however, was actually the 28th regiment 'Regiment orange-nassau' in the Netherlands army, and had a Dutch style uniform. The 2nd (light) regiment, the regiment orange-nassau and the volunteer company were part of perponchers division (A dutch one).
On another note, General Chassé, whose division pushed back the middle guard and saved the British flank from an attack, had commanded the Holland brigade in Spain, against the British and was very anti-British, and made no secret of it.
Edit: Did a little check: the 1st regiment had their own (black) cockade, the others the orange cockade, and thus should be seen as a part of the Dutch army.
