The Netherlands are still officially a Republic in the 1760s, even though the combined offices of Stadtholder and Prince of Orange strongly resemble a hereditary monarchy by this point. If Parliament perceives a conflict of interest between William's role as King of Great Britain and as Stadtholder of the Netherlands, they might try to pressure him to step down as Stadtholder. The Act of Settlement allows them to forbid him to leave the British Isles, and power of the purse is firmly in Parliament's hands, so they've got a strong hand if they decide to press the point.
On the other hand, I rather suspect that in this scenario the Fourth Anglo-Dutch war would be butterflied away. IOTL, it was driven by Dutch support for the American revolutionaries. With William V (*) in place of George III, I could see Britain taking a more conciliatory policy with the Americans and averting or at least significantly delaying outright rebellion.
(*) He's also be William V of Britain, conveniently enough, if William, Duke of Cumberland succeeds George II.