How do Jehovah's Witnesses explain Paul's apparent equating of Yahweh and Jesus in Romans 10?

This isn't a JW only type of thing, for anyone can differentiate which Lord is which in Romans 10, specifically verses 9 to 13 - as do other verses for various other person(s) were called "Lord" (Kyrios).

First you have to realize that Kyrios means "Lord". It relates to relates to Sir, Master, Owner, etc. (technically the status of someone).

In this case, Kyrios does refer to Jesus, however, it also does refer to God the Father as well (both God and his Son are Lords), as do others - for scriptural facts prove this also.

David was the Lord (Kyrios) of Israel. So we see, for example, that God was the Lord of Israel and David was the Lord of Israel (1 Kings 1:36). The authority structure with God and Jesus is exactly the same. God made David the Lord of Israel and now God has made David's son Jesus the Lord of all the world. We would not think that David is God the Father because God had made him Lord.

Abraham is the Lord of Sarah - Sarah laughed to herself, saying, "After I have grown old, and my Lord is old, shall I have pleasure?" (Genesis 18:12; look at 1 Peter 3:9). Sarah obeyed Abraham, calling him "Lord" (1 Peter 3:6).

The two angels came to Sodom in the evening and Lot was sitting in the gate of Sodom. When Lot saw them, he arose to meet them, and bowed himself with his face to the earth, and said, "My Lords, turn aside, I pray you, to your servant's house and spend the night, and wash your feet; then you may rise up early and go on your way."

A Philippian Jailer spoke to these two, and said "Lords, Paul and Silas, what must I do to be saved? (Acts 16:31). There are numerous people in scripture who are called "Lord (Kyrios)." As we can see the word Kyrios isn't exclusive to Jesus or God the Father - but anyone can identify which "Lord" is which between the two.

In Romans 10:13, "Lord" is directly referring to God the Father, not Jesus, for a couple verses before we already see that Jesus is called Lord, specifically verse 9. As the one who saves is indeed the Father, for his grace by means of his Son allows forgiveness of Sin.

Also know that God the Father has made Jesus, the Son, Lord (Kyrios):

The word "Lord" is a term which refer to authority. When Jesus rose from the dead he said, "All authority in heaven and upon the earth has been given to me" (Matthew 28:18). At Acts 2:36 we read that God made Jesus Lord when He raised him from the dead.

Act 2:36 - Let all the house of Israel therefore know for certain that God has made him both Lord and Christ, this Jesus whom you crucified.” (see Matthew 28:18, John 3:35, and Acts 5:31)

Therefore being exalted to the right hand of God, and having received from the Father the promise of the Holy Spirit, He poured out this which you now see and hear. For David did not ascend into the heavens, but he says himself: 'The LORD said to my Lord, "Sit at My right hand, until I make your enemies Your footstool. (Psalms 110:1, see Romans 8:34, Hebrews 8:1, 12:2 and Ephesians 1:20)" ' Therefore let all the house of Israel know assuredly that God has made this Jesus, whom you crucified, both Lord and Christ."

God highly exalted Him, and bestowed on Him the name which is above every name, so that at the name of Jesus every kneel will bow, of those who are in heaven and on earth and under the earth, and that every tongue will confess that Jesus Christ is Lord, to the glory of God the Father. (Philippians 2:9-11).

So Romans 10:9-13 when read (I am using the ESV also)

(9) because, if you confess with your mouth that Jesus is Lord [A] and believe in your heart that God raised him from the dead [B], you will be saved. (10) For with the heart one believes and is justified, and with the mouth one confesses and is saved. (11) For the Scripture says, “Everyone who believes in him will not be put to shame. [C]” (12) For there is no distinction between Jew and Greek; for the same Lord is Lord of all, bestowing his riches on all who call on him. (13) For “everyone who calls on the name of the Lord will be saved [D].”

[A]

Acts 16:31 - And they said, “Believe in the Lord Jesus, and you will be saved, you and your household.”

1 Corinthians 9:6 - Or is it only Barnabas and I who have no right to refrain from working for a living?

2 Corinthians 4:13 - Since we have the same spirit of faith according to what has been written, “I believed, and so I spoke,” we also believe, and so we also speak,

Hebrews 13:15 - Through him then let us continually offer up a sacrifice of praise to God, that is, the fruit of lips that acknowledge his name.

[B]

In scripture, mainly the Greek New Testament, speaks of the Risen Christ and or the Resurrection of Christ - Firstborn out of the Dead, the First-fruits , for such is important when knowing who the Risen Christ what is resurrection even means. It also speaks about God the Father being the one responsible for raising the Christ out of Hades, out of the pangs of death. [C]

Isaiah 28:16 - therefore thus says the Lord God, “Behold, I am the one who has laid as a foundation in Zion, a stone, a tested stone, a precious cornerstone, of a sure foundation: ‘Whoever believes will not be in haste.’

Romans 9:33 - as it is written, “Behold, I am laying in Zion a stone of stumbling, and a rock of offense; and whoever believes in him will not be put to shame.”

[D]

Notes: Some try to prove that Jesus is Jehovah (Yahweh) by pointing to Romans 10:13 and Joel 2:32. The argument is made that Paul is here calling Jesus “Lord”. Since this is a reference to Joel 2:32, where it tells us that whoever calls on the name of Jehovah (Yahweh) will be delivered. Some translations render Romans 10:13 as calling upon Jehovah or Yahweh, in other translation, the name is replaced with LORD/Lord. Jesus (or Yeshua) is the means that Jehovah/Yahweh has provided for salvation (John 3:16,17), no one can come to Jehovah/Yahweh but through Jesus (John 14:6), and no other means has been given by Jehovah/Yahweh for salvation than the name of Jesus. (Acts 4:12).

Acts 2:21 - And it shall come to pass that everyone who calls upon the name of the Lord [Jehovah/Yahweh] shall be saved.

Other examples would be:

Acts 2:23 and 24 - (23) this Jesus, delivered up according to the definite plan and foreknowledge of God, you crucified and killed by the hands of lawless men. (24) God raised him [Jesus] up, loosing the pangs of death, because it was not possible for him [Jesus] to be held by it.

Acts 10:40-43 - (40) but God raised him [Jesus] on the third day and made him [Jesus] to appear, (41) not to all the people but to us who had been chosen by God as witnesses, who ate and drank with him [Jesus] after he [Jesus] rose from the dead. (42) And he [Jesus] commanded us to preach to the people and to testify that he [Jesus] is the one appointed by God to be judge of the living and the dead. (43) To him [Jesus] all the prophets bear witness that everyone who believes in him [Jesus] receives forgiveness of sins through His [God] name.”

For "through His name" is compared to 3 verses, Isaiah 53:11, Jeremiah 31:34 and Daniel 9:24. For believing in the Christ enables one to receive the gift of God's grace by means of his Son - the free and full forgiveness of sin(s) through Christ, which I have mentioned moments ago in this same comment.

As for manuscripts, it was originally YHWH/YHVH (Yahweh/Yah or modern day, Jehovah/Jah). LORD/Lord wherever the Tetragramaton is found was replaced with Adonai, which roughly translates to Lord, for Jews at the time consider God's name very sacred and to honor it they just put Adonai/LORD in its place. Most translations will have LORD/Lord wherever God's name is found, but some translations have put back the God's name into the scriptures - this includes the New World Translation, translated by Jehovah's Witnesses, as do other bible translations who did the same thing in restoring the name in the correct places.

Jehovah's Witnesses believe that Jesus was created, but that he was essentially the very first thing that God created and became his instrument for all sorts of divine activity. They view Jesus as a kind of imprint of God's nature, where the imprint is clearly "divine" (adjective) but not "divine" (noun), just like a coin pressed into clay will bear the image of the coin but not be the coin. Since, according to their theology, God is working through Jesus, they can still occasionally speak about Jesus like you would speak about God, despite the fact that they don't recognize Jesus as literally God.

The explanation of any group or individual regarding which 'Lord' (or Lords) Romans chapter 10 deals with depends upon their theology, as Ben Mordecai has actually pointed out. Any group or individual believing that Jesus was created by God the Father (Jehovah, to Jehovah's Witnesses) cannot see Romans chapter 10 in any way other than, that, although Jesus is the Lord of verse 9, it is the name of Jehovah that must be called upon in order to be saved by verse 13.

However, for those who believe from the Bible that prior to becoming the man, Jesus, he was the eternal, uncreated Word of God who made everything that was made (John 1:1-14) Romans chapter 10 is clear that the Hebraic phrase to "call upon the name of the Lord" (or Yahweh) to be saved (as in Joel 2:32) becomes calling upon the name of Jesus to be saved in the Christian Greek scriptures (Acts 2:38-39). Jesus is the one Lord Christians acclaim. They do not have two Lords (1 Corinthians 8:6).

For Jehovah's Witnesses, when Romans 10:9 says "That if you confess with your mouth, 'Jesus is Lord', and believe in your heart that God raised him from the dead, you will be saved" they agree that that Lord is Jesus, but when verse 13 quotes Joel 3:32, they say THAT Lord is Jehovah and it's actually faith in the name of Jehovah that saves people, not faith in the name of Jesus. That's because, to them, Jesus is subordinate to Jehovah due to having been created by Jehovah.

Now, this is not the place to examine whether the ancient Christian Greek scriptures had the Hebrew letters for Yahweh in verse 13, which Jehovah's Witnesses claim as their reason for refusing to equate Jesus with Yahweh here. Unsurprisingly, their Greek Interlinear footnotes for chapter 10 claim some 'J' manuscript references for putting 'Jehovah' there in place of 'Lord,' whilst insisting that 'Lord' was in ancient texts for verse 9, adding, "Not 'Jehovah'." (Pages 722 & 723 of the Kingdom Interlinear of the Christian Greek Scriptures produced by New World Bible Translation Committee, 1985 edition)

To examine whether the Jehovah's Witness Bible is justified in putting 'Jehovah' in verse 13 yet retaining 'Lord' in verse 9, is another question entirely. This answer simply explains why Jehovah's Witnesses do not equate the Lord of verse 9 with the Lord of verse 13. They do not believe it is theologically possible so to do, and they can give their reasons for that. This question should not provide a platform for Jehovah's Witnesses trying to convince others that their interpretation and translation is correct. The information here is merely being used in this answer to show why Romans chapter 10 presents no conundrum to Jehovah's Witnesses, because they think they can change the Lord in verse 13 to Jehovah and so maintain their clear differentiation between Jesus and Jehovah, which their theology demands.