I am not an expert on NT Wright but I scanned one of his chapters on Justification purely out the interest that your question has aroused. I am Reformed in my view.
All quotes are from NT Wright's 2014 book What Saint Paul Really Said: Was Paul of Tarsus the Real Founder of Christianity?.
His basic reasoning appears to be like this.
First he thinks understanding the Jewish mind at the time of Paul is key to understanding Paul. Then he wants to redefine the word "justification"
'Justification' is a law-court term, and in its Jewish context it
refers to the greatest lawsuit of all: that which will take place on
the great day when the true God judges all the nations, more
particularly the nations that have been oppressing Israel. God will,
at last, find in favour of his people: he will judge the pagan nations
and rescue his true people. Justification' thus describes the coming
great act of redemption and salvation, seen from the point of view of
the covenant (Israel is God's people) on the one hand and the law
court on the other (God's final judgment will be like a great
law-court scene, with Israel winning the case). Learning to 'see' an
event in terms of two great themes like these is part of learning how
first-century Jews understood the world.
In other words it's not about individual justification but national justification.
With this in mind he then wants to join justification with eschatology.
Put these two (justification and eschatology) together, and what
happens? 'Justification', the great moment of salvation seen in terms
of the fulfilment of the covenant and in terms of the last great
law-court scene, would thus also be eschatological: it would be the
final fulfilment of Israel's long-cherished hope. Putting it another
way, the Jewish eschatological hope was hope for justification, for
God to vindicate his people at last.
In other words, he reinforces the idea that Justification is not an individual thing but again, he reasserts it is a national justification afforded to the people of God.
He talks around a few other points in various parts of scripture where Paul is trying to help integrate the Jews and Gentiles under the new gospel. His main goal is to argue that this collective view of group justification, or rather simply being declared a genuine member of God’s people is really what Paul is all about and not about personal justification, having a personal relationship with God, or entering into salvation by faith.
In his concluding remarks he makes the summary around Romans 9:30:
Thus (to follow the train of thought from verse 9:30 onwards), while
Gentiles are discovering covenant membership, characterized by faith,
Israel, clinging to the Torah which defined covenant membership, did
not attain to the Torah. She was determined to have her covenant
membership demarcated by works of Torah, that is, by the things that
kept that membership confined to Jews and Jews only; and, as a result,
she did not submit to God's covenant purposes, his righteousness
(10:3f.); for Christ is the end or goal of the law, so that all who
believe may receive covenant of his people. But 'the gospel' is not an
account of how people get saved. It is, as we saw in an earlier
chapter, the proclamation of the lordship of Jesus Christ. If we could
only get that clear in current debates, a lot of other false
antitheses, not least in thinking about the mission of the church,
would quietly unravel before our eyes. Let us be quite clear. 'The
gospel' is the announcement of Jesus' lordship, which works with power
to bring people into the family of Abraham, now redefined around Jesus
Christ and characterized solely by faith in him. 'Justification' is
the doctrine which insists that all those who have this faith belong
as full members of this family, on this basis and no other.
My impression of the argument is that he simply tries to invert the logic of Paul with no real compelling reason:
Firstly, he seems to neglect the real problem of sin and the terrors of conscience that much greater men and greater thinkers of the past have fled to Christ for refuge. Men like Luther who found a much more persuasive view of Paul’s theology.
Secondly, by assigning such glory to become a member of God’s people as something so much more than receiving the riches of Christ through an imputation of his righteousness, something Paul clearly argued, he seems to insinuate a preposterous idea: that being considered a member of God’s church is able to comfort a sinner individually through identification of the collective body. This is pure nonsense to a great and mighty sinner like me.
I get the feeling the author of the book simply has never sufficiently felt the dread of the Law of God and never really felt how incredible a cleansed joyful conscience apart from works of the law can be. He does not seem to understand the reformers' hearts. He sets himself up, not standing on the shoulders of giants but standing much lower, on a little stool of his own ideas, criticizing the great men of God with unconvincing trifles about collective membership and futuristic justifications of those bodies.
Why he dislikes the individual salvation so much, I will never understand. By emphasizing the collective he seems to actually draw attention away from Christ, whom the sinner so desperately needs individually.
Basically in summary, as a Reformed believer, I take this as another book of hog-wash. I prefer to read Luther.