I found an interesting hypothesis that could actually explain the reason of this apparent discordance among the gospels. I go a bit into the Greek language, so I apologize for that, since this post would be more appropriate for hermeneutics (in fact I would suggest to move there).
In John 19:14 the KJV says that
it was the preparation of the passover, and about the sixth hour
Now, the word in the original translated here as "hour" is the Greek word hora (ὥρα), which is many times translated in this way in the NT.
The interesting thing that I found is that this is not the only possible translation for this word. Here follows a quotation from Thayer's Greek Lexicon, regarding G5610 - hora:
ὥρα, ὥρας, ἡ, from Homer down, the Sept. for עֵת and in Daniel for
2 . the daytime (bounded by the rising and the setting of the sun), a day: ὥρα παρῆλθεν, Matthew 14:15; ἤδη ὥρας πολλῆς γενομένης (or
γινομένης) (A. V. when the day was now far spent), Mark 6:35 (see
πολύς, c. (but note that in the example from Polybius there cited
πολλῆς ὥρας means early)); ὀψίας (ὀψέ T Tr marginal reading WH text)
ἤδη οὔσης τῆς ὥρας (WH marginal reading brackets τῆς ὥρας), Mark 11:11
(ὀψέ τῆς ὥρας, Polybius 3, 83, 7; τῆς ὥρας ἐγιγνετο ὀψέ, Demosthenes,
p. 541, 28).
So the word hora could also be translated as day or daytime, bounded by the rising and the setting of the sun.
To make sense with this possible translation, the word hosei (ὡσεί) which is translated here as "about", should be fixed too. According to Thayer's Greek Lexicon, this word (G5616) can have the following possible translation:
b. about, nearly: α. before numerals: Matthew 14:21; Luke 1:56 (R G);
; John 6:10 (R G L (others ὡς)); Acts 2:41; Acts 4:4 (R G); (in L T Tr
WH it is strengthened here by the addition of περί); ; also, Rec. in
Mark 6:44; R G in John 4:6; John 19:14 (G?), 39; Acts 5:36; Lachmann
in John 6:19 (Judges 3:29; Nehemiah 7:66; Xenophon, Hell. 1, 2, 9; 2,
4, 25). β. before a measure of space: ὡσεί λίθου βολήν, Luke 22:41.
So, when the word hosei is before a numeral, it could be translated as "nearly".
And this is exactly the case, since the word hosei, in John 19:14 (TR), is right before the numeral hectos (sixth):
ωρα δε ωσει εκτη
(hora de hosei hectos)
So, if we consider this possible alternative translation, the verse would sound something like this:
It was the preparation of the Passover, and nearly the sixth daytime
Now the sixth daytime sounds much like Friday morning in the Hebrew culture.
So it could also be translated as:
It was the preparation of the Passover, and nearly the sixth day's
Now, there is a good reason why somebody (as the apostle John) would want to specify that it was nearly the morning of the sixth day, and it is that Friday was normally called the Preparation (see Mark 15:42 and Luke 23:54) and the day before the Passover was also called preparation. So, to give a right chronology of the facts, it would have been important to specify that it was the preparation of the Passover AND the preparation of the Sabbath, or in other words the sixth day. John may have chosen to say that it was nearly the sixth day's morning to imply both that it was Friday and also that it was early in the morning, as the other gospels say.
So, according to what I found, it could easily be that when Jesus was presented before the Jews by Pilate with the words, "Behold your king", it was nearly Friday morning and then, at the third hour Jesus was crucified (around 9 am).
This would harmonize the account of Jesus' crucifixion in all the four gospels, placing it on Friday morning.