In general, political campaign material in the UK is not formally regulated, and it is a matter for the press and public to decide on the basis of such material whether they consider it reasonable and accurate.
While the Advertising Standards Authority (ASA) is the UK's independent regulator for advertising across all media, political adverts are not within its remit. The ASA suggests that anyone with concerns about a political advert could contact the party responsible, and exercise their democratic right to tell them what they think. The free press also has an important role in holding politicians and campaigners to account.
There has been a concern following the EU referendum with regards to the accuracy of information fed to the public. In principle, I agree completely that information disseminated during election and referendum campaigns should be as accurate as possible; it is incumbent on politicians to ensure that this is the case and for their opponent to hold them to account for anything that may be construed as misleading. However, in practice, this proves far more difficult, as data and statistics will always be interpreted in different ways by different people, especially in the pursuit of a specific outcome. Even then, in reality, there are very few occasions when politicians wilfully misinform the public.
Throughout the referendum campaign there was a demand for accurate projections as to how the UK would be affected in the future, for both sides of the debate. However, future predictions cannot be infallible, no matter how much each side believes in the strength of its argument and veracity of its sources. Of course, there were opportunities to fact-check particular statements and there are a large number of websites which run their own service doing just this. Even then, it can prove fiendishly difficult to say with absolute certainty whether an argument is indeed factually inaccurate and should, therefore be completely disregarded. Fundamentally, it is in the hands of the public to make this distinction.
Of course, the wider law does generally apply to political campaign material, including the law of defamation and public order offences. Electoral law does enforce regulation in terms of ensuring parties and other campaigners to include an 'imprint' on their campaign material, identifying its source, to ensure transparency and accountability. Electoral law also makes it a criminal offence to publish false statements about a candidate: the courts do enforce this legislation, as illustrated during the April 2015 election court ruling which disqualified the mayor of Tower Hamlets for a litany of illegal practices.