Under European copyright law, the copyright on recorded music lasted 50 years since a song or an album is first released, while the songwriter's publishing copyright lasts 70 years after the composer's death.
Since "Love Me Do" and its B-side, "P.S. I Love You," were released in 1962, protection for the tracks expired on December 31st, 2012.
To prevent many songs from slipping into the public domain, in the 2000's Cliff Richard, the British pop icon, and others campaigned for a copyright extensions for performers who performed on a sound recording. Then, the European Union passed the so-called "Sir Cliff's Law" i.e. the EC Directive 2011/77, granting an extension from 50 to 70 years of the term of protection for a sound recording.
Such law hasn't yet come into effect since the changes are still being implemented into the laws of each member state across the European Union. The process of implementation will set to be complete by November of this year.
Then, for everything recorded before 1963, they technically fall within the public domain in Europe: the copyright for the song remains, but the licensing fees usually provided to publishers has lapsed, meaning that anyone can distribute the original recordings without permission for the record labels.
In another twist, the EU Term Extension conditions include a 'use it or lose it' clause for anything recorded before 1963, so that labels will have to cede control over its copyright to performers, if it does not market the sound recording containing the performance. Labels and performers shall prove that they were still invested in bringing them to new audiences in the following years.
Then, Sony Music and Legacy recordings are doing that for Bob Dylan's song, since the recording copyright are beginning to expire. For instance, Sony released late last year a very limited, 100-copy, of 50th Anniversary Collection box set, including 86 unreleased Dylan tracks dating back to 1962 and 1963 - or, better said, 50 and 51 years.