Was Christ born on December 25, 1 BC or December 25, AD 1? 

Is the question about calculating calendar years or about Jesus Himself?

Either way, this article might spell it out: https://www.ncregister.com/blog/what-year-was-jesus-born-the-answer-may-surprise-you (To be clear, not recommending or endorsing the author or the publication.)

Curiosity, more than anything.  Thanks for the link; I thought the Church might have had an official teaching, but it looks as though all we have are various opinions.

I would say neither.

When Christ was 1 year old, wouldn't it be the year 1 AD?

I've seen various dates and time lines, all within about a hundred years or so of each other, but for whatever reason the one that sticks in my head is that Christ was born in the year 5199, i.e. 5199 years after creation.

There was no year zero, so wouldn’t it necessarily have to be either 1 BC or 1 AD?

He was born on 25 December 1 BC.

He turned one year old on 25 December AD 1.

Ya got me lol
But I don't see how it could be 1 BC, and if it was 1 AD, then what was the year when Christ was 1 year old? 2 AD? Doesn't make any sense to me. Saying He was born in the year 5199 after Creation makes sense to me.

Christ would most likely have been born in 2 BC. So He would be 1 year old on Dec 25, 1 BC. He would have been 2 years old on Dec 25, 1 AD. Continuing in the same fashion, He would be 34 years old on Dec 25, 33 AD. But He was crucified in that year around April 3rd, 33 AD, which the Jews call Nisan 14 in their calendar as the day of the Passover/Pasch. So, on April 3rd, 33 AD, Christ's 34th birthday not yet having come around, He would have died around the age of 33 years and 3 months. Therefore, a 2 BC date seems more likely than a 1 BC date imo. Anyway, 2033 is the 2000th anniversary of His crucifixion and resurrection and God may well have a surprise for humanity on that date. Let's pray and prepare to celebrate the 2000th anniversary of Our Lord's crucifixion/resurrection. We now 33 AD is the date, and not 30 AD as some modernists think, because St. Luke tells us John the Baptist began preaching in the 15th year of the Emperor Tiberius, and that is 29 AD. After 3 1/2 years of ministry, Christ died, was buried and then rose from the dead. So the AD 33 date works and the AD 30 date does not work for the resurrection. Luke also tells us Jesus Christ was around 30 years of age in 29 AD, which also fits in with a 2 BC date for His birth.