One example in my personal experience that has always made me think about this was a set of identical twin brothers I knew in High School. They both grew up in a stable, two parent, Christian home. From somewhere along in the 4th or 5th grade, it was quite obvious one was homosexual and one was heterosexual.
In my thinking, if it were nature (purely genetic), homosexuals would have disappeared in one generation because they can't reproduce. My knowledge of the twins above argues strongly against nurture. What does that leave? Genetic mutation? Combination of nature and nurture?
Throughout history, most gay people married. Oscar Wilde was married to Constance Lloyd, and they had two sons, Cyril (1885) and Vyvyan (1886). The gay Episocopal bishop in New Hampshire, the Rev. Gene Robinson, was married and likewise had two children with his wife before he ditched wife and kids and went off to live with his boyfriend. New Jersey governor Jim McGreeey married twice and had children with both of his wives, before he famously announced he was gay.
While many lesbian mothers have them through artificial insemination or adoption, I think it's true that most lesbian "parent" couples involve a partner who is a divorcee or who became an unwed mother the usual way --- with a man --- and whose children later get adopted by the lesbian partner.
So homosexuals can and do marry someone of the opposite sex, and can and do reproduce. Always have. It's just that they still have a sexual yen for a person of the same sex.
And why? Like you said, it's maybe part nature, part nurture, and often a wholly mysterious thing, even to themselves.