I don't think you should place too much emphasis on this particular act, which in any case became invalid with the Restoration in 1660 The new Parliament was pretty quick to pass something similar with the similarly titled Navigation Act of 1660.
Both Acts however were just two in a sequence of measures designed to implement mercantilist economic ideas, and it would be better to think of the influence of mercantilism rather than just one instance of it. It would also be wrong to think that only the British were up to this kind of thing.
With regard to short-term local consequences the Act was certainly provocative to the Netherlands and helped lead to war between the Commonwealth and the Dutch. (As, eventually, succeeding acts in the same vein would help lead to the breakaway of the American colonies.)