My understanding is that under the legal framework in the U.K., the personal data, such as names of people who have signed petitions, can be kept only as long as is necessary, and then has to be destroyed. So any petition will remain open for six months, and then the text of the petition and the number of signatories will be recorded in the House's official papers. But I think soon after that, the names of the signatories would likely be destroyed, because they are no longer necessary, so there is no legal basis to keep them.
On December 9th, 2014. See this statement in context.