Short answer : Not a lot.

Long answer : If they reach a certain amount, they get discussed in parliament and you get an official response to the issue. The response is normally "no". I don't know of any occasion of one causing major changes.

What it does do though, is force (somebody within) the government to take a position on an issue, and they can no longer say they are ignorant of it. Having an official response means they must explain their counterarguments and reason for denial. If this is then brought up by other means in future, you know what you're arguing against.

Depending on where you live and who your local MP is, you may get better results writing to your local MP, who in theory should be supporting the views of their constituents.

Ultimately, it's a bit weak. Perhaps the response is worth it, as a stepping stone to further methods on an issue - though it's been suggested that it's a distraction, making people fill in a form to "get the anger out of their system", and then not protesting or raising awareness by other means.

I still fill them in, but I don't expect much from them.