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].”
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.
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.”
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
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.