By the time he wrote his letter to the Romans Paul was back.  He abandoned all pretense of being concerned about nothing among you except Jesus Christ, and him crucified.¹  His letter began with a history lesson like those he preached at Pisidian Antioch and Athens.  It was the history of sin since the flood.  For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of people who suppress the truth by their unrighteousness…²

I think it is important not to confuse ungodliness here with sinful acts.  The word is ἀσέβειαν (a form of ἀσέβεια) in Greek from ἀσεβής, a compound word, the negation of σέβομαι (a form of σέβω; “to worship, honour, revere”).  This ἀσέβεια is something in people that causes them not to revere God in truth.  What I am calling the religious mind is surely part of this ἀσέβεια (ungodliness).  Because of this ἀσέβεια people either ignored or denied the obvious truths that were made plain to them by the creation itself, because what can be known about God is plain to them, because God has made it plain to them.  For since the creation of the world his invisible attributes – his eternal power (δύναμις) and divine nature – have been clearly seen, because they are understood through what has been made.  So people are without excuse.³

So people are culpable for their failure to worship Him in truth, and his wrath was visited upon them as they worshiped after their own inclinations (not unlike Nadab and Abihu in some cases).  For although they knew God, they did not glorify him as God or give him thanks,⁴ but they became futile in their thoughts and their senseless hearts were darkened.  Although they claimed to be wise, they became fools and exchanged the glory of the immortal God for an image resembling mortal human beings or birds or four-footed animals or reptiles.⁵

In other words both people and the rest of creation were made in such a way by God that it should have been plain to us that no human being, bird, mammal or reptile made everything that had been made.  But this ἀσέβεια in human beings makes our thoughts futile, our hearts senseless, and though it may flatter us that we are wise, ultimately it makes fools of us before God.  And so God’s wrath against our ἀσέβειαν (a form of ἀσέβεια) and our unrighteous worship was revealed (Ἀποκαλύπτεται, a form of ἀποκαλύπτω) from heaven.

The desires of their hearts were not necessarily evil because of the Greek word ἐπιθυμίαις (a form of ἐπιθυμία).  Jesus said, I have earnestly (ἐπιθυμίᾳ, another form of ἐπιθυμία) desired (ἐπεθύμησα, a form of ἐπιθυμέω) to eat this Passover with you before I suffer.⁷  But ἀσέβεια (ungodliness) even in religious people nudges those desires toward evil.  You people are from your father the devil, Jesus said to those who were trying to kill him,⁸ and you want to do what your father desires (ἐπιθυμίας, a form of ἐπιθυμία).  He was a murderer from the beginning…⁹

So the idea here is that when God gave those who exchanged the glory of the immortal God for an image resembling mortal human beings or birds or four-footed animals or reptiles over to impurity (ἀκαθαρσίαν, a form of ἀκαθαρσία), this was their own desire (ἐπιθυμίαις τῶν καρδιῶν, “desire of the heart”).  In other words, He stopped restraining their urges.  Later, Paul wrote the believers in Ephesus (Ephesians 4:17-19 NET):

So I say this, and insist in the Lord, that you no longer live as the Gentiles¹⁰ do, in the futility of their thinking.  They are darkened in their understanding, being alienated from the life of God because of the ignorance that is in them due to the hardness of their hearts [Table].  Because they are callous, they have given themselves over (παρέδωκαν, a form of παραδίδωμι) to indecency for the practice of every kind of impurity (ἀκαθαρσίας, a form of ἀκαθαρσία) with greediness.

The Greek word ἀκαθαρσίας (a form of ἀκαθαρσία) comes from the word ἀκάθαρτος, (unclean), used extensively in the Gospels to describe unclean¹¹ spirits.  I mean, Paul wrote the Corinthians, that what the pagans¹² sacrifice¹³ is to demons and not to God.¹⁴  Paul was not the originator of this explanation of the gods, he paraphrased Moses’ prophecy to Israel: They made him jealous with other gods, they enraged him with abhorrent idols.  They sacrificed to demons, not God, to gods they had not known; to new gods who had recently come along, gods your ancestors had not known about.¹⁵

So God gave them—who exchanged the glory of the immortal God for an image resembling mortal human beings or birds or four-footed animals or reptiles—over in the desires of their hearts to the impurity of demon worship, to dishonor (ἀτιμάζεσθαι, a form of ἀτιμάζω) their bodies (σώματα, a form of σῶμα) among themselves.  The word ἀτιμάζω is from ἄτιμος, the negation of τιμή.  Paul wrote the Thessalonians (1 Thessalonians 4:3-5 NET):

For this is God’s will (θέλημα): that you become holy, that you keep away from sexual immorality (πορνείας, a form of πορνεία), that each of you know how to possess his own body in holiness and honor (τιμῇ, a form of τιμή), not in lustful (ἐπιθυμίας, another form of ἐπιθυμία) passion (πάθει, a form of πάθος) like the Gentiles who do not know (εἰδότα, a form of εἴδω) God.

These Gentiles did not know (εἰδότα, a form of εἴδω) or see God because they were not born from above.  I tell you the solemn truth, Jesus explained to Nicodemus, unless a person is born from above, he cannot see (ἰδεῖν, another form of εἴδω) the kingdom of God.¹⁶  So I know the cure, or antidote, Paul proposed to this ἀσέβειαν (ἀσέβεια, ungodliness) that caused the Gentiles not to εἰδότα (εἴδω) God.  And I am beginning to get a fairly clear picture of how the demon worshippers dishonored their bodies among themselves.  They were in this fix because they did not glorify (ἐδόξασαν, a form of δοξάζω) him as God or give him thanks.  Paul wrote the Corinthians (1 Corinthians 6:18-20 NET Table):

Flee sexual immorality (πορνείαν, another form of πορνεία)! “Every sin a person commits is outside of the body (σώματος, another form of σῶμα)” – but the immoral person (πορνεύων, a form of πορνεύω) sins against his own body (σῶμα).  Or do you not know that your body (σῶμα) is the temple of the Holy Spirit who is in you, whom you have from God, and you are not your own?  For you were bought at a price.  Therefore glorify (δοξάσατε, another form of δοξάζω) God with your body (σώματι, another form of σῶμα).

I’m convinced the ἀτιμάζεσθαι (ἀτιμάζω) Paul referred to was πορνείας (πορνεία), the cultic worship practices that even descendants of Israel followed before (and after) they were liberated from Egypt.  (And as I mentioned before I hope that Paul used the word πορνείας in 1 Thessalonians 4:3 to describe adultery rather than cultic worship practices.)  So where my religious mind may have conceived of God’s wrath as punishment (even as atonement) for sin, Paul’s Christ mind conceived God’s wrath as God releasing people to even more sinfulness—the desires of their hearts—because it is God who makes people righteous, not people themselves.

Addendum: April 21, 2020
Tables comparing Deuteronomy 32:16 and 32:17 in the Tanakh, KJV and NET, and Deuteronomy 32:16 and 32:17 in the Septuagint (BLB and Elpenor) follow.  Following those are tables comparing Romans 1:21; 1:24; Ephesians 4:17; 1 Corinthians 10:20 and John 3:3 in the NET and KJV.