Just checked my 1958, 1961 and 1965 Handbooks and they all have AFN as 0500-0005 GMT. Next one I have is 1972 and that has them as 24 hours.
In 1958 Norddeutscher Rundfunk has an overnight programme on 971 Thursday, Friday, Saturday and Sunday also carried by Westdeutscher Rundfunk on the same channel.
RIAS Berlin is 24 hours on 683 as is Deutschlandsender East Germany on 728 and 782.
Rome 845 is 24 hours with the Notturno dall'Italia programme 2235-0540 including newscasts in Italian, French, English and German.
Can't spot anything else overnight in the European section.