This can turn out to be quite a large question and so I will address the occurrence of the three terms only. I find none of the three exact terms, "New Israelites", "New Israel" or "Spiritual Israel", in the pages of Scripture per se.
That being said, Jesus did declare His intention to gather sheep that were not currently part of "this fold" and make one flock:
And other sheep I have, which are not of this fold: them also I must bring, and they shall hear my voice; and there shall be one fold, and one shepherd. - John 10:16
In Romans 9, Paul states quite clearly that natural descent within the nation of Israel does not guarantee inclusion within what should be considered the people of God's promise:
I say the truth in Christ, I lie not, my conscience also bearing me
witness in the Holy Ghost, That I have great heaviness and continual
sorrow in my heart. For I could wish that myself were accursed from
Christ for my brethren, my kinsmen according to the flesh: Who are
Israelites; to whom pertaineth the adoption, and the glory, and the
covenants, and the giving of the law, and the service of God, and the
promises; Whose are the fathers, and of whom as concerning the flesh
Christ came, who is over all, God blessed for ever. Amen. Not as
though the word of God hath taken none effect. For they are not all
Israel, which are of Israel: Neither, because they are the seed of
Abraham, are they all children: but, In Isaac shall thy seed be
called. That is, They which are the children of the flesh, these are
not the children of God: but the children of the promise are counted
for the seed. - Romans 9:1-8
Later in that same epistle Paul explains that Gentiles, rather than replacing something or constituting something altogether new, have been grafted in to something which already exists:
I say then, Have they stumbled that they should fall? God forbid: but rather through their fall salvation is come unto the Gentiles, for to provoke them to jealousy. Now if the fall of them be the riches of the world, and the diminishing of them the riches of the Gentiles; how much more their fulness? For I speak to you Gentiles, inasmuch as I am the apostle of the Gentiles, I magnify mine office: If by any means I may provoke to emulation them which are my flesh, and might save some of them. For if the casting away of them be the reconciling of the world, what shall the receiving of them be, but life from the dead? For if the firstfruit be holy, the lump is also holy: and if the root be holy, so are the branches. And if some of the branches be broken off, and thou, being a wild olive tree, wert graffed in among them, and with them partakest of the root and fatness of the olive tree;... - Romans 11:11-17
Galatians 6 contains a reference to the Israel of God which, though sometimes a source of contention, is contextually a reference to those who are in Christ from either Jew or Gentile stock and who walk according to the rule that the Old Covenant profits nothing:
As many as desire to make a fair shew in the flesh, they constrain you to be circumcised; only lest they should suffer persecution for the cross of Christ. For neither they themselves who are circumcised keep the law; but desire to have you circumcised, that they may glory in your flesh. But God forbid that I should glory, save in the cross of our Lord Jesus Christ, by whom the world is crucified unto me, and I unto the world. For in Christ Jesus neither circumcision availeth any thing, nor uncircumcision, but a new creature. And as many as walk according to this rule, peace be on them, and mercy, and upon the Israel of God. - Galatians 6:12-16
There is a remnant of (natural) Israel, elected (according to foreknowledge and grace) unto the promise of Abraham, and there are Gentiles elected by that same grace and these are one in Christ. There are not two, but one; neither slave nor free, male nor female, Jew nor gentile.
What appears unto us a replacement or distinction is just that; an appearance to us. God knows those who are his. The Shepherd knows His sheep and calls them by name from every tribe and tongue and people. Those who hear His voice and follow are Abraham's children and heirs according to promise:
Know ye therefore that they which are of faith, the same are the children of Abraham. - Galatians 3:7
Not something new or the replacement of something old but that which was always intended ... Christ Jesus, the Israel of God. Israel should not rely on lineage and Gentiles should not boast on being grafted in.
Boast not against the branches. But if thou boast, thou bearest not the root, but the root thee. Thou wilt say then, The branches were broken off, that I might be graffed in. Well; because of unbelief they were broken off, and thou standest by faith. Be not highminded, but fear: For if God spared not the natural branches, take heed lest he also spare not thee. Behold therefore the goodness and severity of God: on them which fell, severity; but toward thee, goodness, if thou continue in his goodness: otherwise thou also shalt be cut off. And they also, if they abide not still in unbelief, shall be graffed in: for God is able to graff them in again. For if thou wert cut out of the olive tree which is wild by nature, and wert graffed contrary to nature into a good olive tree: how much more shall these, which be the natural branches, be graffed into their own olive tree? 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. - Romans 11:18-27
In this way, all Israel will be saved.
