The short answer
Historians who have gone looking for Smith's fingerprints on actual legislation have mostly come back disappointed. One of them turns the causation round: the policy came first, and the book's standing followed from it.
In Parliament
The first recorded quotation of the book in the House of Commons was Charles James Fox's, in 1783, for a household platitude about living within your income. He later told a friend he had never read it.
William Pitt's famous tribute of 1792 praised it for explaining how capital accumulates, not for free trade. In the great free-trade fight of 1846 both sides quoted Smith. In 1849 a former Lord Chancellor defended the Navigation Acts out of Smith as well, calling him my illustrious master. The Acts were repealed anyway.
As an adviser
The best evidence is in his letters. He wrote memoranda for ministers on the American war, on free trade for Ireland, and in 1780 on customs. In 1786 the minister he had written to, William Eden, signed a commercial treaty with France that cut duties on both sides. Whether the book caused it is still argued; the sequence is not.
What it did change
It settled what an argument about trade had to look like. After Smith, a case for a monopoly could not be made in the old terms of treasure and the balance of trade; it had to answer his question, cheaper than what, and paid for by whom. And one proposition of his, that the division of labour is limited by the extent of the market, is still doing work in economics.