Stephen Sizer illustrates the continuity between the OT people of God (Israel) and the NT people of God (the church):-

Israel: The Church in the Old Testament	The Body of Christ: The Church in the New Testament
Righteous live by faithfulness (Hab 2:4)	Righteous live by faith (Rom 1:17)
Holy people (Deut 7:6; 33:3; Num 16:3)	Holy people (Eph 1:1; Rom 1:7)
Chosen (Deut 7:6; 14:2)	Chosen (Col 3:12; Titus 1:1)
Called (Isa 41:9; 2 Chron 7:14)	Called (Rom 1:6-7; 1 Cor 1:2)
‘Church’ = Assembly in Greek (Mic 2:5)	Church (Mt 16:18; 18:17; Eph 2:20)
Flock (Eze 34:2, 7; Psalm 77:20)	Flock (Lk 12:32; Acts 20:28)
Holy nation (Ex 19:6)	Holy nation (1 Pet 2:9)
Treasured possession & kingdom of priests (Ex 19:5-6)	Special possession & royal priesthood (1 Pet 2:9)
Children of God and People of God (Hos 1:10; 2:23)	Children of God (Jn 1:12); People of God ( 1 Pet 2:10)
People of his inheritance (Deut 4:20)	Glorious inheritance (Eph 1:18)
My dwelling place = tabernacle (Lev 26:11; Eze 37:27)	Dwelling among us = tabernacle (Jn 1:14; 2 Cor 6:16)
God is a husband betrothed ( Isa 54:5; Jer 3:14; Hos 2:19)	Christ is a husband betrothed (2 Cor 11:2; Eph 5:25-30)
Twelve tribes (Gen 49:28; Rev 21:12)	Twelve Apostles (Mk 3:14; Rev 21:14)

As Wayne Grudem (Systematic Theology, pp859-863) explains, Evangelical Protestants have held different views on the relationship between Israel and the church.

Dispensationalists such as Lewis Sperry Chafer hold that God has two distinct plans for the two groups.  God’s purposes for Israel are for earthly blessings, and will be fulfilled on this earth at some point in the future.  His plans for the church are for heavenly blessings, and these will be fulfilled in heaven.  The distinction will be especially apparent in the millennium, when Israel will reign on earth as God’s people, whereas the church will have already been taken up to heaven at the time of the rapture.

‘Progressive’ dispensationalists deny that Israel and the church are on different tracks with regard to God’s purposes, and make no distinction between them in the future eternal state.  However, they still maintain that the Old Testament prophecies concerning Israel will be fulfilled in the millennium by ethnic Jewish followers of Jesus Christ.  Therefore, they deny that the church is the ‘new Israel’ and they deny that the Old Testament prophecies are fulfilled in the church.

Protestant and Catholic theology generally teaches that the church includes both Old Testament and New Testament believers.  Many non-dispensationalists believe that there will be a future large-scale conversion of the Jewish people, Rom 11:12-31.  But this conversion will result in Jewish believers becoming part of the one church of God – they will be “grafted back into their own olive tree”, Rom 11:24.

Many New Testament passages refer to the church as the ‘new Israel’ or new ‘people of God’.

Paul distinguishes between a person who is a Jew because of physical descent from Abraham and by physical circumcision, and a ‘real Jew’ who is one inwardly, with a circumcision of the heart, Romans 2:28f.

Similarly, Abraham himself is not to be considered only as father of the Jewish people in a physical sense.  He is also father of all who believe, without being circumcised, and who follow his example, Rom 4:11-12.

So, says, Paul, ‘not all who are descended from Israel belong to Israel, and not all are children of Abraham because they are his descendent…it is nt the children of the flesh who are the children of God, but the children of the promise are reckoned as descendants, Rom 9:6-8.

It is, accordingly, the church which is now God’s chosen people.  Gal 3:29 and Phil 3:3 teach the same thing.

Far from thinking of the church as a separate group from the Jewish people, Paul writes to Gentile believers at Ephesus telling them that they have been made one with Jewish believers.  They are ‘fellow citizens with the saints and members of the household of God,’ Eph 2:12-20.

Hebrews 8:8-10 quotes Jeremiah 31:31-34, which refers to God’s new covenant with the house of Israel and with the house of Judah, and says that this is the covenant that has now been made with the church.

James writes his general letter to Christian churches and refers to them as ‘the twelve tribes in the Dispersion’, James 1:1.

Peter writes to New Testament Christians as ‘exiles of the dispersion’, 1 Peter 1:1, and in 1 Pet 2:4-10 he teaches that God has bestowed on the church so many blessings that were promised to Israel in the Old Testament: Christians are the new ‘temple’; and a ‘royal priesthood’; they are the ‘chosen race’, a ‘holy nation’, and ‘God’s people’.

‘What further statement could be needed in order for us to say with assurance that the church has now become the true Israel of God and will receive all the blessings promised to Israel in the Old Testament?’

Steve Motyer (art. ‘Israel’ in Evangelical Dictionary of Biblical Theology) argues similarly:-

The Holy Spirit has been poured out equally on Jews and Gentiles alike.  For Paul, the clear implication of this is that God has abolished the distinction between them.  Both alike are offered salvation on the same terms, Rom 3:27–30; 10:11–13; Eph 2:11–22).

Although he preached the gospel to the Jews first, Rom 1:16; cf. Acts 13:46, he was under obligation to all, Rom 1:14–15.

Has God has repudiated his commitment to Israel?  No, says Paul.  God’s word has not failed, Rom 9:6.  But the covenant terms which had previously applied to Israel are now applied to all believers in the mixed congregations to whom he wrote – Jews and Gentiles alike.  They are

• God’s elect,1 Thess 1:4–5
• a holy people, 1 Cor 1:2
• a justified people, 1 Cor 6:11; Rom 3:22–24
• the redeemed, Rom 3:24; Eph 1:7
• those who inherit the kingdom of God, 1 Cor 6:10, Col 1:12
• the children of God, Rom. 8:14; cf. Exod. 4:22
• those on whom the glory of God rests, Rom 5:2; 8:30
• those who offer pleasing worship, Rom 12:1–2; Eph 5:1–2
• those who can  appeal to the covenant faithfulness of God, now revealed in Christ, Rom 8:31–39

It is very probable that ‘the Israel of God’, in Galatians 6:16, is the church.

Paul refers to Abraham as universal father, Gal 3; Rom 4. He argues that the fulfillment of God’s promise to Abraham must mean that he is a source of blessing, not only for Israel, but for the whole world.  Those who belong to Jesus are the seed of Abraham, and heirs of the promise, Gal 3:16, 29. Unbelieving Israel are like fruitless branches cut off from the olive tree (but still capable of being grafted back on), whereas believing Gentiles are like wild shoots that have been grafted on, Rom 11:17–18.