It's not your responsibility to cut their hedge and if you choose to cut the overgrowth on your side you should offer the cuttings to your neighbour before disposing.
Different councils have different rules. My council don't offer a free service for private gardens, East Riding Council do if you are elderly or disabled but only if you are a council tenant living in adapted or sheltered housing.
Check your council's website. There is a possibility they provide this service but as the hedge belongs to your neighbour it is highly unlikely.
No harm in asking, though