Gen 49.10 is a rather cryptic passage, and it's not obvious how to interpret it.
If it is talking about a physical government, the first king was Saul, from Gibeah of the tribe of Benjamin and not from Judah. David was from Judah, it's true, but with the Babylonian captivity the scepter clearly departed to gentile nations, first Babylonia, then Persia, Greece, the Seleucids and only briefly returned with the Hasmoneans who were not of the line of Judah (but they were at least jewish), and then power fell to the Romans.
So something else must be meant by this passage when it is referring to "sceptre".
Here is a discussion of some interpretations in Theological Wordbook of the Old Testament:
Gen 49:10, Jacob’s blessing on his son Judah: “The staff shall not
depart from Judah, nor the sceptre from between his feet until
‘Shiloh’ come, and the obedience of the people be unto him.” What is
meant by the phrase “until Shiloh come”? If it is a prophecy of Jesus,
the NT nowhere alludes to it. It would be preferable to say it is a
prophecy of David as a type of the Messiah, but what does Shiloh add’?
What are the other possibilities of interpretation?
One is to connect
consonantal šylh with the rare Akkadian word šēlu “counselor.”
second is to retain the translation Shiloh and read either, as in KJV,
“Until Shiloh come(s)” but what can this possibly mean?
“Until he (Judah/David) comes to Shiloh,” expressing some expansion of
the base of power beyond Judaean borders.
A third suggestion is made
on the almost unanimous testimony of the ancient versions (LXX and
Targums) that would infer a slight change in the mt, i.e. a change of
šylh to šlh (= šellô “to whom it belongs”), the relative particle š,
the preposition l, the third person singular pronominal suffix -ô. The
phrase then could read “until he comes into his own.” Allegro notes
the messianic interpretation of Gen 49:10 at Qumran where the verse is
applied to the scion of David to come (see Bibliography). The
similarity to Ezk 21:27 [H 32] is striking.
A fourth suggestion is to
divide the consonantal text šylh into šay lô “until tribute is brought
to him” (Moran, Speiser).
Hamilton, V. P. (1999). 2376 שִׁילֹה. R. L. Harris, G. L. Archer Jr., & B. K. Waltke (Eds.), Theological Wordbook of the Old Testament (electronic ed., p. 919). Chicago: Moody Press.
For details on the Messianic interpretation at Qumran, the cited work is online at http://johnallegro.org/text/Further_Messianic_References_in_Qumran_Literature.pdf