This depends on the test you apply to a given petition to decide if it's resulted in a policy change of the type you're searching for.
Has a government ever directly attributed its policy changes to a petition on Parliament's website? Not to my knowledge. But this test is a distortion of the purpose of petitions in general.
The indirect goal of a petition might be to change a policy, but its more direct objective is to produce quantifiable evidence of the strength of public opinion in favour of that change. Petitioners have reasonable hope that this will indirectly accomplish the greater goal because governments do care about public opinion, and so a successful petition may contribute to a chain of events:
- The government sees the petition and perceives that public opinion is stronger than if nobody had bothered to create or sign one.
- The government perceives that opposing public opinion on this issue will be harmful to their electoral chances in future.
- The government is more sympathetic to a change of policy because future electoral success always features highly on their list of priorities.
Even setting aside the question of the role of petitions in assessing public opinion, it's very difficult to find a government making any acknowledgement of the influence of public opinion on their decision. They will always prefer to claim that the new policy is the natural conclusion of the policy they had all along, or that they thought very carefully on principled grounds and arrived at the same conclusion as the public, or that the facts changed and the new policy is now the obvious choice. But it's universally understood that politicians make their decisions on more self-serving grounds than they let on, and pleasing a soon-to-be-voting-again public is one of those grounds.
With this in mind, the closest we can really get to testing the effectiveness of a petition is whether the government eventually changed its policy in the petition's favour. We will never know if the same decisions would have been made without the petition, but we can also say for certain that there was some reflection about public opinion involved (because the same is true for every political decision) and be reasonably confident that the petition makes some sort of difference to the government's perception of that public opinion.
So which petitions have preceded a government policy change in their favour? I thumbed through the full list of 59 petitions that have been debated in Parliament and there are three that I think can be said to meet this test:
Additionally, Make Orkambi available on the NHS for people with Cystic Fibrosis is the subject of an ongoing parliamentary inquiry so could join this list in the course of time.
An incredible number of the remaining are either Brexit-related demands (generally either completely opposing each other or duplicating similar demands) or request things that aren't obviously opposed to existing Government policy or that are outside the Government's remit. Yet by my count there are still about 20 petitions out of the remaining 53 with specific demands that have been given a firm "no" by the Government. Petitions which are followed by change in their favour are still well in the minority.
But this is in the nature of petitions. While they can play a role in illustrating the weight of public opinion when government policy shows resistance to yielding to it, if the electoral pressure is too intense it's in the interests of governments to get out ahead of that public opinion and buy an easy win; and there are plenty of other ways to keep tabs on public opinion. Petitions naturally occupy the margin of ideas where it's arguable exactly how much public support there is for an idea and so there's an incentive for its adherents to demonstrate they exist in meaningful numbers. Occasionally this probably does have some effect in politics.