The sixties was a period of great cultural change in Britain, which affected attitudes towards illegitimacy as well as many other social and sexual mores. The attitudes towards illegitimacy varied with class, age, religious faith, and, of course, individual opinion. In general, there was a generational shift with illegitimacy becoming more common with younger people, a trend that has continued to the point where now almost half of all children in Britain are born outside conventional marriage. Two other trends were that the lower the social class, the lower the education level, and the less religious a woman, the more likely that she would have a child outside of marriage.
The so-called "sexual revolution" of the 1960s, which permeated popular culture and is reflected in much of the music of the period, involved both an attitude in favour of sexuality being free and spontaneous (although there was still a double standard, treating men as "studs" and women as "sluts" for identical sexual behaviour) and an enabling technology of the (widely accessible for the first time) birth control pill.
Perhaps this is best expressed in Philip Larkin's poem:
Sexual intercourse began
In nineteen sixty-three
(which was rather late for me) -
Between the end of the "Chatterley" ban
And the Beatles' first LP.
Up to then there'd only been
A sort of bargaining,
A wrangle for the ring,
A shame that started at sixteen
And spread to everything.
Then all at once the quarrel sank:
Everyone felt the same,
And every life became
A brilliant breaking of the bank,
A quite unlosable game. ...