I think in my lifetime (all in the US), I’ve heard them pronounced as “nine-teen oh one,” etc. However, in my youth, having known people who were alive during those times, they (at least some of them) omitted the “oh,” so it would just be “nine-teen one,” etc.
What would have been the more common pronunciation, with or without “oh?”
Another common reading would have been “nineteen aught one”, where “aught” meant “zero”.
Not really an answer, but an anecdote:
Somewhere at home, I’ve got an 1899 or 1900 college yearbook. There was space in the book devoted to each class, and some rather breezy, unofficial writing. The classes were referred to as “the Naughty-naughts,” “the Naughty-ones,” “the Naughty-twos,” etc. I suspect that was more tongue-in-cheek naughtiness (heh) than actual serious usage, but it amused me.
Another vote for this. When I was a kid, I was around quite a few people who were old enough to remember those years, but I don’t remember them saying “nineteen”.
They would say, “back in aught one” (1901) or “I left home in aught eight” (1908)…
One of the most famous rifle cartridges ever invented and still in widespread use is the 30.06 invented in 1906. It is called the thirty aught six.
I’m gonna say without:
-I distinctly recall an announcer on a recorded radio program dating from the late 30s ask, “Do you remember back in nineteen eight and nine…”
-My grandpa always told people he had been born in nineteen two (which he had been).
I now call the years in this decade “aughts” as in aught seven. Just because of the strange looks I get.
My dad, who died in 1990, always said he was born in “aught six.”
It freaks me out to think that my dad’s centenary was two years ago.
I believe that my dad referred to his birthdate as “nineteen ‘o’ six.” I may have heard some of his contemporaries use “nineteen ‘and’ six,” etc. Teachers may have preferred that version. I don’t think some of the teachers liked a zero referred to as an o.
I will add this, and then, someone more versed in “The Simpsons” will come along after and fix the quote for me.
As spoken by Abe Simpson (Homer’s Father):
"…that was back in Nineteen Dickety 2. Of course back then they took our word for Twenty, so we had to use 'Dickety. "
I’m not sure what word / phrase we will use, but the current contenders are light years ahead of some other choices outher there. 
If I could chose, I’d go with Aught. – I don’t think we will ever “offically” get a phrase however.
I don’t think we will use “aught”. “Aught” and “Naught” are antiquated words that I don’t see young people ever using…
Damn whippersnappers.
“Now my story begins in 19-dickety-two. We had to say ‘dickety’ cause the Kaiser had stolen our word ‘twenty’. I chased that rascal to get it back, but gave up after dickety-six miles.”
I think my father (born also in '06) said 19 oh 6. Maybe 19 6 too, but never 19 aught 6. FWIW, he was born and grew up in South Philly.
I’m certain that for most of my life, my relations and teachers have referred to “Nineteen-one” and so on, all the way up to “Nineteen-nine”, and the same thing for previous centuries, such as the Battle of Trafalgar in Eighteen-five. I’m pretty sure it was commonplace here in England to do so, up to the Millennium, where it was plain that “twenty-one” would not do as the name of a year.
To my mind, this rule of inserting “oh” need not be applied to older dates where no ambiguity exists, so I continue to tell people that my house was built in nineteen-one, despite the questioning looks I get in return.
I think it’s a matter of historical record that the “oh” was first put into numbers when telephones came into use. My grandparents had what’s known these days as a candlestick phone, with no dial. You lifted the receiver and then told the operator what number you wanted. When you answered, you would confirm your own number: “Hastings two-nine-oh-three” instead of saying “hello” or your own name.
The “oh” in dates was certainly used in Britain, at any rate by the military. In The Life and Death of Colonel Blimp (1943), Major-General Clive Wynne-Candy speaks of “nineteen-oh-nine”. To my ears it sounds like precise military jargon, along with “oh-two hundred hours” meaning two o’clock.
I always listen out for others who like me don’t cave in to popular fashion on this point, and salute them, for we are a dying breed. The most recent I heard on the radio was Katherine Whitehorn, journalist, aged eighty-six.
Perhaps what part of the world a person ‘hails from’ has something to do with it, also? 
Now we’re getting into what zombies call it. I reiterate that my father was born in nineteen oh six and he would have grown up without any telephones, since his family was very poor.
Exactly. Zoe’s Dad is from Ireland.
I have occasionally heard the form of “Nineteen and five”.