I'm not aware that the Navy's ever merged or anything to that extent, they were still two separate nations - now with a connection. However, there was a much larger degree of cooperation and collaboration between the nations. The revolutionary Dutch Merchant system was exported to Britain, and joint ventures or peaceable settlements were more common. An example of this is the division of Asia between the Dutch and British East India Companies, the Dutch got the spice islands and the British got India. The cooperation meant both nations could specialise on one trade and maximise profits in that region, while not being in competition with each other (it's worth noting this was an agreement primarily between chartered companies and not Government policy).

Basically, the Navy's didn't merge as they were still separate countries, but there was a large element of cooperation between merchants in Britain and The Netherlands, which allowed both to benefit.

Source : The Silk Road by Peter Frankopan