He send out a message saying all captains should surrender their ships to the British and all colonies should co-orporate with the British to fight the French and the rebels
Nobody listened, of course (Even though many naval captains were loyal to him, the crews really weren't), but the British took this as an welcome excuse to attack the colonies. They also used it as an excuse to keep the Cape in 1815; 'Yeah your dad gave it to us in twenty years ago, so we are gonna keep it now".
During the 1799 invasion however, quite a few ships surrendered themselves to the British and became an independent 'Dutch' squadron within the Royal navy, under the flag of the Prince of Orange. There was also a Holland Brigade made out of prisoners of war and orangists, which was disbanded in 1802.. The ships were sold to the Royal navy.
On the case of Orangeism:
William (Future king I) made a deal with Napoleon that made him Prince of Nassau-Orange-Fulda in 1803, betrayed him in 1806, and was kicked out of power. He entered Austrian service in 1809 and got wounded at Wagram. It's important to remember that the Oranges had already given up all hope to the Dutch throne until he was suddenly called back. The fact he returned in 1813 was not because the people wanted it, but because it was the only way to ensure the victorious Allies wouldn't punish the Netherlands (Some smart politicians at work there). Though, of course, our history books say otherwise.
Yeah, I edited this post like 5 times.