When using a resourcepack that replaces the records, if you enter a menu (e.g. inventory), the records will only play the default sound. If the record is already in your hotbar when you enter the world it will play the correct song from the resourcepack as long as you don't enter a menu.

EDIT: In the 1.7.1 Prerelease, the records will play either the default sound or the custom one; it's random, but entering menus no longer affect it.