No, you have the legal obligation to maintain the trees and hedges. They are your hedges.
It is not uncommon for councils to trim such plants that overhang onto public rights of way but they are under obligation to do so. They will sometimes do so only after the owner has been asked to remove offending branches and hasn't - in which case they may try and charge.
As for the height for privacy, it depends how tall you are; there is no requirement to constrain a hedge to a height though extreme examples that cause nuisance to neighbours may be the subject of action under the Statutory Nuisance legislation.