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This is a very difficult question to answer, not because the scriptures aren't clear but because so many people in our generation are living in the sin of adultery due to an unlawful remarriage.
You referenced 1 Cor. 7:15 in your OP but maybe you should take a second look at verse 10 in the same chapter.
"To the married I give this command (not I, but the Lord): A wife must not separate from her husband. 11 But if she does, she must remain unmarried or else be reconciled to her husband. And a husband must not divorce his wife."
Paraphrasing the commandments of Jesus himself, Paul teaches us that there are only two options for a divorced believer regardless of the reason for divorce or who files for divorce. Remain unmarried or be reconciled.
Quoting: DeputyDog That is to the church..meaning two believers
Two problems with that view point:
1) Paul does not make the same distinction you're making. His message is to the "married" and he makes no distinctions concerning whether both spouses are believers or only one of them.
2) The believing spouse in v. 15 obviously falls under the admonition to the believers in v. 10 not to remarry after a divorce. Do you think he is allowing remarriage for some believers but not others?
Paul is talking to the church. Two believers because he says not I but the Lord, and he gives instructions to BOTH the husband and the wife. You cannot give commands to an unbeliever. He then goes on in verse 12 saying to THE REST I say this...making a distinction from verse 10
To “the rest” Paul (as an inspired apostle whom the Lord counted trustworthy and who had the Spirit of God, v. 25, 40) applies Matthew 19:6, 9 and explains that marriage between a Christian and an unbeliever is indeed a legitimate marriage. At no time in this passage does the inspired apostle urge the Christian who is married to an unbeliever to depart (divorce) because his or her mate is an unbeliever. Indeed, the Lord's will is that they remain together as Matthew 19:5-6 instructs.
So, if the unbeliever is content or willing to live with a Christian, the Christian is commanded not to depart, for the Christian brings a sanctifying influence and life into the home (v. 14; cf. 1 Pet. 3:1-2).
However, if the unbeliever departs (we necessarily infer that under such a condition the unbeliever is not content to dwell with the Christian), the Christian is “not under bondage” to the unbeliever (the Christian is to let the unbeliever depart). Can we know with confidence what “not under bondage” means? Yes we can, by applying the same type of hermeneutics we apply to other Bible passages (including those already mentioned in this study).
These principles include, for instance:
1) The meaning given the passage will not contradict other passages of scripture (Jno. 17:17).
2) All God says on the subject must be consulted (Psa. 119:160).
3) The context (both immediate and general) must be observed and harmonized.
4) The proper definition of terms is vital in coming to a correct understanding of the text. (We cannot assume definitions - we must verify their legitimate meaning and usage in scripture).
5) The grammar used by the Holy Spirit must be considered and respected (cf. Matt. 22:31-32).
Now, when we correctly apply these principles of interpretation to 1 Corinthians 7:15, it will not be men's scholarship on the passage or men's division over the passage which determines our understanding and our application of it. It will be because we have “rightly divided” the word of truth and come to “understand what the will of the Lord is” (2 Tim. 2:15; Eph. 3:3-4; 5:17).
