How long had John the Baptist been preaching before Jesus started his public ministry?

The answer to the OP question about how long did John the Baptist preach before Jesus' ministry began is about 6 months.

John was the son of Zacharias and Elizabeth. Zacharias was of the tribe of Levi.

There was in the days of Herod, the king of Judaea, a certain priest named Zacharias, of the course of Abia: and his wife was of the daughters of Aaron, and her name was Elisabeth. Luke 1:5

Priests began their ministry at age 30.

From thirty years old and upward even until fifty years old, all that enter into the host, to do the work in the tabernacle of the congregation. Num 4:3

So, we know John the Baptist began his ministry at age 30.

John was born 6 months before Jesus.

And in the sixth month [of Elizabeth's pregnancy] the angel Gabriel was sent from God unto a city of Galilee, named Nazareth, [to Mary] Luke 1:6

John is born. At age 30, he begins his ministry of baptizing. Jesus comes to him about Jesus' age of 30.

Now when all the people were baptized, it came to pass, that Jesus also being baptized, and praying, the heaven was opened, And the Holy Ghost descended in a bodily shape like a dove upon him, and a voice came from heaven, which said, Thou art my beloved Son; in thee I am well pleased. And Jesus himself began to be about thirty years of age, being (as was supposed) the son of Joseph, which was the son of Heli, Luke 3:21-23

The phrase "began to be about thirty years of age" means what it says. Jesus wasn't about the age of 33, but 30. Here are translations.

Now Jesus himself was about thirty years old when he began his ministry. He was the son, so it was thought, of Joseph, the son of Heli, (NIV)

So Jesus, when he began his ministry, was about thirty years old. He was the son (as was supposed) of Joseph, the son of Heli, (NET)

Jesus, when he began his ministry, was about thirty years of age, being the son (as was supposed) of Joseph, the son of Heli, (RSV)

Jesus, when he began his ministry, was about thirty years of age, being the son (as was supposed) of Joseph, the son of Heli, (ESV)

So, John had been preaching about 6 months after his age of 30 and then baptized Jesus at about Jesus' age 30.

PS There is a claim of no coins from that period showing the co-regency, but here is one from this answer elsewhere

or here

besides here is a list of imminent scholars using the co-regency.

PPS additionally, there is a claim about Roman historians and a co-regency.

Tacitus a Roman historian (emphasis mine)

Tiberius Nero and Claudius Drusus, his stepsons, he honoured with imperial tides, although his own family was as yet undiminished. ... [Tiberius] Nero remained alone of the stepsons, and in him everything tended to centre. He was adopted as a son, as a colleague in empire and a partner [with Augustus] in the tribunitian power, and paraded through all the armies, no longer through his mother's secret intrigues, but at her open suggestion. Tacitus Anal I.3

Suetonius Roman historian (emphasis mine)

XXI. Since the consuls caused a law to be passed soon after this that he [Tiberius] should govern the provinces jointly with Augustus and hold the census with him, he set out for Illyricum on the conclusion of the lustral ceremonies; but he was at once recalled, and finding Augustus in his last illness but still alive, he spent an entire day with him in private. Suetonius Lives of the Caesars XXI

So, again to answer the OP, the written evidence points to 6 months between when John the Baptist began his ministry and when he baptized Jesus, which then marked the beginning of Jesus' ministry.

To reinforce this, the Aaronic priesthood began the same way. Moses washed Aaron. Thus they could minister to the LORD (Ex 40:12-13).

Summary conclusion

You could say our Lord's public ministry started with his baptism. But note that there were 40 days in the wilderness after his baptism. But his baptism was the ordination for his public ministry, so I will focus on our Lord’s baptism.

The wedding of Cana, which might be taken as the beginning of his public self-revelation began about 50 days after his baptism, and the Passover of 7th April AD 30 was at most about 60 days after his baptism. This would mean our Lord was baptised about 5th February AD 30. It seems unreasonable to me to suppose that there was much delay in being baptised after reaching 30 years of age. If we allow a month’s delay - that there might be certainty of reaching 30, in order to be certain to fulfil all righteousness by keeping the whole law of Moses (Numbers 4:3) - then this would lead to our Lord being born about 1st January 1 BC.

The difficulty lies with correctly interpreting Luke 3:1. John's ministry could have started anything up to nearly a full year and ten months before our Lord’s baptism.

Our Lord's baptism, must have begun after reaching 30 years of age, because his ministry was priestly (Numbers 4:3) making the sacrifice of himself, as well as a prophetic.

But John's ministry was not priestly but merely prophetic. So there is no good reason to believe John began his ministry after reaching 30 years of age; and so there is also no reason to suppose his ministry began just six months before the baptism of our Lord.

Evidences our Lord was born (very) early 1 BC or (very) late 2 BC

If we assume that the lunar eclipse mentioned by Josephus is the one of 10th January 1 BC then it turns out that Luke 3:23 and the account in John 1:19 to John 2:13 make a remarkably tight time frame for both the birth and the baptism of our Lord, upon very reasonable assumptions.

Josephus records that Herod the Great died after a lunar eclipse and before the following Passover. At the end of the nineteenth century Emil Schurer argued that the eclipse being referred to happened on the 13th March 4 BC, (a partial eclipse). Andrew Steinmann and others argue that the number of days between this eclipse and the Passover, 29 days, is not nearly enough for all the events recorded by Josephus as happening between the two events. They argue even if the twelve events spoken of by Josephus were to take a minimum amount of time they would still have taken at least 41 days.

They thus argue that the eclipse being referred to was the (full) lunar eclipse of 10th January 1 BC, where there are enough days, 89 days, between the eclipse and Passover for all the events mentioned by Josephus. This is fairly explained on Wikipedia under Herod the Great, see section on death.

I agree with Hold to the Rod that our Lord was crucified 3rd April AD 33 (Julian Date). There are a lot of reasons for accepting this date. The date held as the majority view of scholars for the last hundred year, because of Schurer, has been AD 30. The AD 33 date strongly supports the view Herod died between 10 Jan 1 BC and the Passover, 89 days later.

John 1:19 to John 2:13

Luke's Gospel tells us our Lord was about 30 years old when either his ministry began or he was baptised (Luke 3:23). Seeing that his baptism is best seen, at least in part, as the ordination for his ministry (see Luke 3:22) then it must be he was at least and about 30 at his baptism for his priestly work (Numbers 4:3).

If we suppose Jesus was born exactly 1st January 1 BC, just prior to the lunar eclipse preceding Herod’s death, then he would have been 30 years old on 1st January AD 30.

There is a breathless speed in John's gospel from John 1:19 right up to the Passover of John 2:13. (John 1:29, 35, 43, 2:1 (it took 3 days max to reach Cana), wedding (1 or 2 days?), a few days in Capernaum (John 2:12), and then “the Jews’ Passover was at hand” (John 2:13). Assuming 2 days at the wedding, five days at Capernaum , a three day journey to Jerusalem, and an arrival 2 days before Passover, there are a total of 18 days from John 1:19 to the Passover.

Steinmann is surely correct in saying our Lord had already been baptised and spent 40 days tempted before John 1:19. He then returned to John the Baptist at Bethabara (John 1:28) where let us suppose he spent a couple of days interacting with all. So we have baptism 42 days before John 1:19. So there were about 60 days between his baptism and Passover.

The Passover of AD 30 fell on 7th April ("Babylonian Chronology" by Richard Parker & Dubberstein, 1956). From this we can surmise that Lord was baptised on about 5th February AD 30.

We can also surmise there was no need to delay his baptism much more than a month after his 30th birthday. Why delay? The only need to delay was to make sure he kept the Mosaic law of Numbers 4:3. It follows our Lord was probably born about 1st January 1 BC, not much later than the 5th January 1 BC, or maybe a week or so earlier.

Luke 3:1 - the 15th year of Tiberius - possible meanings

In all the options of interpretation below, I assume (from above) that our Lord was baptised about 5nd February AD 30.

Option A: If Tiberius's 15th year began on 17th September AD 28 – which was the day, in AD 14, the Senate appointed Tiberius (having been permitted by Tiberius) then John's ministry could have lasted up to a year and four and a half months before our Lord Jesus' baptism.

Note prior to options B & C: Steinmann draws attention to what Ernest Martin has noted; that surviving coins and inscriptions reckon Tiberius' reign from either 1st January AD 15 (the beginning of the Roman regnal year) or 19th August AD 14, the death of Augustus (Steinmann, note on page 219, referencing page 89 of "The Nativity and Herod's Death" by Ernest Martin, 1989).

Option B: If we take Luke 3:1 to mean 19th August AD 28 as the beginning of Tiberius' 15th year, then up to nearly a year and five and a half months are possible for the ministry of John before the baptism of our Lord.

Option C: If we take Luke 3:1 to mean 1st January AD 29 as the beginning of Tiberius’ 15th year, then up to one year and a month is possible for the ministry of John before our Lord’s baptism.

Option D: There is a final possibility. If Luke is writing from a Jewish perspective then he might be using the following Jewish convention: the period from August (AD 14) or September AD 14 up to 1 Nisan, (about April) AD 15, is Tiberius' 1st year, and from 1 Nisan AD 15 to 1 Nisan AD 16 is his second year, etc. This would mean that Tiberius 15th year began on 1st Nisan AD 28, 15th April (Parker & Dubberstein). This would mean that John's ministry could have been anything up to a year and nine and a half months before the baptism of our Lord.

In summary, all that can thus be said is that John’s ministry began anything up to a year and nine and a half months before the baptism of our Lord, but no more.

John's ministry continued a while in tandem with that of our Lord (John 3:23, 24). John 5:35, John "was a burning and a shining light", indicates by this time John was beheaded.