In a comment above, Lucian referred to Justinian's Digest, page 100, paragraph 8 where I read "According to the Roman custom, the day begins at midnight and ends at the middle of the next night." The Wikipedia reference states that the civil day ran from midnight to midnight, while the natural day was from sunrise to sunset.
So, the question is whether John used the Roman civil day or natural day. Matthew, Mark and Luke used the natural day which is also the Jewish system. But John was written in the Greek-Roman city of Ephesus long after the fall of Jerusalem and Jewish dispersion, so he might well have used the Roman civil day. It is no problem to use two systems at the same time. I have lived for many years in Kenya where we have both systems. If you speak English, you can say 9 a.m. and that is translated into Swahili as "the third hour" (saa tatu). So, we are used to adding or subtracting 6 hours when moving from one system to the other.
If does not make sense to assume that John at times used one system and at other times used the other system, so we need to look at the four places where John mentions the hour of the day, that is: 1,39; 4:6, 52 and 19:14.
Alfred Edersheim discusses 4:52 on page 296 in his book: The Life and Times of Jesus, The Messiah.
So he asked them the hour when he began to get better, and they said to him, “Yesterday at the seventh hour the fever left him.” (ESV)
Edersheim explains: "Now, however the Jewish servants may originally have expressed themselves, it seems impossible to assume, that St. John intended any other than the Roman notation of the civil day, or that he meant any other hour than 7 p.m. The opposite view, that it marks Jewish notation of time, or 1 p.m., is beset by almost unsurmountable difficulties."
The problem is that if it was 1 p.m., the father could have reached home to Capernaum from Cana before dark, and he would undoubtedly be eager to see if the son was healed. The servants left home as soon as the son was better, so they should have met about halfway in the middle of the afternoon. But if that were the case, the servants could not possibly have used the word "yesterday". On the other hand, if it had been 7 p.m., the father would have stayed overnight, as it was dangerous to travel at night. The servants would have done the same, so they met in the middle of the morning the next day. Then the servants could have said "Yesterday".
If we look at 1:35-39, we read:
The next day again John was standing with two of his disciples, 36 and he looked at Jesus as he walked by and said, “Behold, the Lamb of God!” 37 The two disciples heard him say this, and they followed Jesus. 38 Jesus turned and saw them following and said to them, “What are you seeking?” And they said to him, “Rabbi” (which means Teacher), “where are you staying?” 39 He said to them, “Come and you will see.” So they came and saw where he was staying, and they stayed with him that day, for it was about the tenth hour. (ESV)
This paragraph starts out with "the next day". That would be in the morning, and the first thing that happened was that they saw Jesus walk by. Two disciples followed Jesus and went with him to where he stayed. This fits nicely with 10 a.m. and with the fact that they stayed with him the rest of that day. The context is not conclusive one way or the other, but 10 a.m. is more likely that 4 p.m.
Then we have 4:3-6
[Jesus] left Judea and departed again for Galilee. 4 And he had to pass through Samaria. 5 So he came to a town of Samaria called Sychar, near the field that Jacob had given to his son Joseph. 6 Jacob's well was there; so Jesus, wearied as he was from his journey, was sitting beside the well. It was about the sixth hour. (ESV)
Jesus was tired from a long day's journey and sat down to rest while the disciples went to find some food in the village. They planned to stay overnight outside as they thought the people to be hostile. This fits much better with about 6 p.m. than about noon. This was the normal time to come for water. The conversation would not take many minutes, and the distance to the town was short. There is no problem for this to have happened in the late hours of the afternoon and early into the evening. Jesus stayed there for two days preaching to them.
The final passage is 19:14:
Now it was Preparation day of the Passover [Feast]. The hour was about the sixth (DLNT)
Jesus was brought to Pilate immediately after sunrise, and after being taken before Pilate, then Herod and back to Pilate, he was brought to Golgotha at 9 a.m. There is ample time for these events as Jerusalem was a small city with a short distance from Pilate to Herod to Piltate to Golgotha just outside the city wall.
We need to recognize that Preparation Day was the standard way of referring to what we call Friday, the day before the Sabbath. It is never in Greek used to refer to the day before a feast day. And the "Passover" here stands for the weeklong Passover Festival. It is not the day when the Passover meal was eaten in the evening. The Greek has a genitive construction meaning Passover's Friday (preparation day for the following Sabbath).
We have earlier seen in 18:28:
Then they lead Jesus from Caiaphas to the Praetorium. And it was early morning. And they themselves did not enter into the Praetorium in order that they might not be defiled, but might eat the Passover [Feast] (DLNT)
To "eat" the Passover Feast simply means to celebrate the weeklong feast, and there was an important meal on the day after the main Passover meal.
Let me end with a quote from Edersheim concerning this topic and recommend for those interested to read the full account.
many writers have interpreted the expression ‘the Passover’ as referring to the Paschal Supper, and have argued that, according to the Fourth Gospel, our Lord did not on the previous evening partake of the Paschal Lamb, or else that in this respect the account of the Fourth Gospel does not accord with that of the Synoptists. But as, for the reason just stated, it is impossible to refer the expression ‘Passover’ to the Paschal Supper, we have only to inquire whether the term is not also applied to other offerings. And here both the Old Testament and Jewish writings show, that the term Pesach, or ‘Passover,’ was applied not only to the Paschal Lamb, but to all the Passover sacrifices, especailly to what was called the Chagigah, or festive offering (from Chag, or Chagag, to bring the festive sacrifice usual at each of the three Great Feasts). According to the express rule (Chag. i. 3) the Chagigah was brought on the first festive Paschal Day. It was offered immediately after the morning-service, and eaten on that day - probably some time before the evening, when, as we shall by-and-by see, another ceremony claimed public attention. We can therefore quite understand that, not on the eve of the Passover but on the first Paschal day, the Sanhedrists would avoid incurring a defilement which, lasting till the evening, would not only have involved them in the inconvenience of Levitical defilement on the first festive day, but have actually prevented their offering on that day the Passover, festive sacrifice, or Chagigah.