Scriptural basis for replacement theology? There exist groups who believe that Christians or believers of Jesus completely replaced Jews as Israel and as the chosen people, and that Jews no longer have a valid relationship with G-d, unless they become Christians.

Is such a theology prevalent within Christianity? What is the scriptural basis both for and against it?
You kind of tipped your hand by using the word ideology in your question! An unscriptural ideology is at the heart of a deficient theology regarding Israel and the Christian Church. (By Church, with a capital C, I'm referring to the universal or &quot;holy catholic&quot; church, which is composed of all true believers in Jesus Christ.)
There Is No Legitimate Argument For It!
The key biblical passage in this regard would have to be Romans, chapters 9-11, where Paul attempts to counteract an incorrect ideology with a correct theology. By the way, if you entertain any thoughts about Paul having slipped into ideology instead of theology, read 9:3-5.

&quot;For I could wish that I myself were accursed, separated from Christ for the sake of my brethren, my kinsmen according to the flesh, who are Israelites, to whom belongs the adoption as sons, and the glory and the covenants and the giving of the Law and the temple service and the promises, whose are the fathers, and from whom is the Christ according to the flesh, who is over all, God blessed forever. Amen&quot; (Italicized words are original to the NASB Updated version).

What Paul is saying, in part, is there would be no Christianity without Judaism. According to Paul, Judaism is the very foundation of Christianity. Most significant, of course, is that Jesus was born a Jew!
Most Evangelical Christians I know do not subscribe to the notion that Israel is washed up, or that God has given up on Israel and subsequently turned to the church as his only light of truth in a dark world. They focus, rather, on the very expressions Paul uses to indicate the temporary nature of Israel being put on hold, so to speak. Paul uses such expressions as

&quot;. . . it is not as though the word of God has failed&quot; (9:6)

In other words, the word of God may not have had the effect God desired, but that is not to say it will never achieve God's desired outcome.

&quot;. . . the children of promise are regarded as descendants&quot; (9:8; notice the tense of the verb I've italicized: present--not past--tense).

Put differently, descent from Abraham, the father of faith and a nation, will on day--once more--be an advantage to those descendants who believe God.
Let's stop here. In Paul's line of reasoning, what God was doing among Israel is simply being continued today through the church until the promises God made to Abraham and Isaac, &quot;the child of promise,&quot; are fulfilled completely.
The Christian Church as Parenthetical
Think of the &quot;church age&quot; as parenthetical. Not in the sense that what is inside the parentheses is unimportant or an afterthought. No, but in the sense that what came before the parentheses is going to be continued after the parentheses. Other expressions Paul uses are also of relevance:

&quot;. . . but Israel , pursuing a law of righteousness, did not arrive at that law . . .. Because they did not pursue it by faith, but as though it were by works. They stumbled over the stumbling stone&quot; (and then Paul quotes Isaiah 8:14). Note: the italicized words are in the the NASB Updated version.

Put differently, Israel stumbled at the simplicity of simply believing in the promise of God; namely, that he was going to bless the entire earth through Abraham's seed (and Isaac's seed, and King David's seed, and so on). Israel got caught up in the rigid observance of the Law as their way of earning God's favor with works of righteousness and stumbled at the sufficiency of faith to win God's favor, which is at the heart of the biblical notion of grace (viz., God's undeserved, unmerited favor to sinners, whether Jew or Gentile).

&quot;. . . there is no distinction between Jew and Greek; for the same Lord is Lord of all, abounding in riches for all who call on Him; for 'WHOEVER WILL CALL ON THE NAME OF THE LORD WILL BE SAVED'&quot; (10:12-13).

Another key verse is the following:

&quot;. . . God has not rejected His people, has He? May it never be! For I . . . [Paul] am an Israelite . . .&quot; (11:1 NASB Updated).
&quot;In the same way then, there has also come to be at the present time a remnant according to God's gracious choice&quot; (ibid., v.5, original italics)
&quot;But by their [Israel's] transgression salvation has come to the Gentiles, to make . . . [Israel] jealous. Now if their transgression is riches for the world and their failure is riches for the Gentiles, how much more will their fulfillment be!&quot; (ibid., v.12, original italics).

In other words, a time of fulfillment is coming for Israel. Again, that fulfillment comes after the parentheses of the church age.
Israel Is the Cultivated Stock; the Gentiles, A Wild Branch Grafted In
Paul also resorts to metaphor in chapter 11, talking of bread dough, trees, and branches. The Jews are the original tree root of the cultivated olive, whereas the Gentiles are the &quot;wild&quot; branches which were grafted into the tree when the original branches were &quot;broken off&quot; (vv.16-17). He then warns the Gentiles not to become arrogant. After all, the Jews came first. Just because some of their branches were broken off doesn't mean the original root can't or won't come back to life and produce more branches from the original root!
Moreover, God can just as easily break off the grafted-in branches (i.e., the Gentiles) for unbelief when they fail to lay hold of God's kindness while it is still offered to them (see vv.18-24).
A Partial and Temporary Hardening
Finally, Paul wraps up his argument by saying,

&quot;. . . a partial hardening has happened to Israel until the fullness of the Gentiles has come in; and so all Israel shall be saved, just as it is written [in Isaiah 59:20].&quot;

Furthermore,

&quot;. . . the gifts and the calling of God [to Abraham, Isaac, and Jacob, and so on] are irrevocable. . . .For God has shut up all in disobedience so that He may show mercy to all&quot; [both Gentile and Jew] (v.32).

In conclusion, God has not replaced Israel with the Christian church; rather, the Christian church was grafted in as a way of continuing what God had been doing all along with Israel (viz., seeking a people for himself who would latch on to his promises and thereby be declared righteous by God through simple faith, just as Abraham's faith was accounted unto him as righteousness (Romans 4:3 and 9), since &quot;the just shall live by faith.&quot;