Will all Israel be saved?
Scripture teaches that Israel as a nation will indeed be saved, but this refers to a remnant chosen by God's grace rather than every single Jewish person, and this turning will happen suddenly, near the end of this age when the Lord returns.
The promise in Romans 11:25-26 is clear: blindness has only come upon Israel in part, and it will last until the fulness of the Gentiles has come in. After that, all Israel shall be saved, for the Deliverer shall come out of Sion and turn away ungodliness from Jacob. This is described as God's own covenant, that He will take away their sins, fulfilling what was spoken through Isaiah that the Lord would set His hand again to recover the remnant of His people from the nations where they were scattered.
This salvation of Israel is tied to God's election of grace, not to the works or belief of the nation as a whole. Romans 11:5 shows that even now there is a remnant according to the election of grace, meaning God has reserved to Himself those He knows will hear and receive the gospel, even in the time known as Jacob's trouble. Israel as a whole has not obtained what it seeks because it does not seek by faith, but the elect among them will obtain it, while the rest remain blinded for a season.
The conversion of this remnant is pictured as happening suddenly, in one day, not through a gradual process. Zechariah 3 speaks of the Lord removing the iniquity of the land in one day, and Micah 7 declares that God will have compassion, subdue their iniquities, and cast their sins into the depths of the sea. This appears to take place at the end of the age when Christ returns, for the Jewish remnant will see Him whom they pierced, mourn deeply as one mourns for a firstborn son, and then receive Him, saying, Blessed is He who comes in the name of the Lord. At that point He will turn away all ungodliness from the nation.
So the remnant that survives the wrath of the antichrist and endures to that day is the group with whom this promise is fulfilled. God has concluded all, both Jew and Gentile, in unbelief, so that He might have mercy upon all. His gifts and calling toward Israel are without repentance, meaning He will not go back on His word, and His love for them as the beloved of the fathers guarantees that He will revisit them in the last days, turning the remnant of the nation to Himself.
The scriptures
For I would not, brethren, that ye should be ignorant of this mystery, lest ye should be wise in your own conceits; that blindness in part is happened to Israel, until the fulness of the Gentiles be come in. And so all Israel shall be saved: as it is written, There shall come out of Sion the Deliverer, and shall turn away ungodliness from Jacob: For this is my covenant unto them, when I shall take away their sins. As concerning the gospel, they are enemies for your sakes: but as touching the election, they are beloved for the fathers’ sakes. For the gifts and calling of God are without repentance. For as ye in times past have not believed God, yet have now obtained mercy through their unbelief: Even so have these also now not believed, that through your mercy they also may obtain mercy. For God hath concluded them all in unbelief, that he might have mercy upon all.
Even so then at this present time also there is a remnant according to the election of grace. And if by grace, then is it no more of works: otherwise grace is no more grace. But if it be of works, then is it no more grace: otherwise work is no more work. What then? Israel hath not obtained that which he seeketh for; but the election hath obtained it, and the rest were blinded.
Hear now, O Joshua the high priest, thou, and thy fellows that sit before thee: for they are men wondered at: for, behold, I will bring forth my servant the BRANCH. For behold the stone that I have laid before Joshua; upon one stone shall be seven eyes: behold, I will engrave the graving thereof, saith the LORD of hosts, and I will remove the iniquity of that land in one day. In that day, saith the LORD of hosts, shall ye call every man his neighbour under the vine and under the fig tree.
Who is a God like unto thee, that pardoneth iniquity, and passeth by the transgression of the remnant of his heritage? he retaineth not his anger for ever, because he delighteth in mercy. He will turn again, he will have compassion upon us; he will subdue our iniquities; and thou wilt cast all their sins into the depths of the sea.
And it shall come to pass in that day, that the Lord shall set his hand again the second time to recover the remnant of his people, which shall be left, from Assyria, and from Egypt, and from Pathros, and from Cush, and from Elam, and from Shinar, and from Hamath, and from the islands of the sea.
And I will pour upon the house of David, and upon the inhabitants of Jerusalem, the spirit of grace and of supplications: and they shall look upon me whom they have pierced, and they shall mourn for him, as one mourneth for his only son, and shall be in bitterness for him, as one that is in bitterness for his firstborn.
Behold, he cometh with clouds; and every eye shall see him, and they also which pierced him: and all kindreds of the earth shall wail because of him. Even so, Amen.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.