Hello! Thank you for your question. It is a very good one.
At Got Questions ministries, we have different writers with slightly differing views on the end times. So that is why there is a bit of difference of opinion as to what will happen.
We understand that God's word is NOT a matter of anyone's interpretation. So some opinions (including my own) can be wrong. However, I will give evidence from the Bible what has been said regarding this issue, and after prayer and using the wisdom God has given me, I write this answer. Not everyone from Got Questions will agree with me. Since this is NOT a MUST BELIEVE issue, it is possible to agree to disagree. But let's let God's word speak for itself.
Articles such as this one
https://www.gotquestions.org/Old-Testament-believers.html
might need to be updated.
First, I believe the Old Testament saints are already in heaven. This is the actual heaven where God's throne is, and not a separate part of Sheol, where people are comforted. The word Sheol simply means the grave.
While there could be a separate part of Sheol where people are comforted, no verse in the Bible actually explicitly says this. I believe all the Old Testament saints are in heaven already. Here is why.
The Bible says:
"Even so Abraham believed God, and it was reckoned to him as righteousness. Therefore, be sure that it is those who are of faith who are sons of Abraham. The Scripture, foreseeing that God would justify the Gentiles by faith, preached the gospel beforehand to Abraham, saying, 'All the nations will be blessed in you.' So then those who are of faith are blessed with Abraham, the believer." (Galatians 3:6-9)
This means:
Abraham was credited with righteousness before the time of Christ. God deemed Abraham righteous, because he believed God's word as it was revealed to him.
God preached a shortened version of the gospel to Abraham (Galatians 3:8).
All who believe by faith, who are born later, are called children of Abraham and blessed with him.
So for Old Testament saints, it can be said, they had something like a credit card. The credit card is their faith in God. The Old Testament saints did not have the full gospel but a partial one. This gospel was slowly revealed to them (whether they understood it or not). If they believed God's word to them (even if they did not see the whole picture), God credited righteousness to them beforehand.
So like a credit card, in which a person gets the goods, but the payment is withdrawn from the bank later, so also Jesus died later for Old Testament saints, but God gave them the goods (heaven) first! Jesus died for people of the past, present and future. He died once and for all.
Abraham, who is called the father of faith, believed in the gospel according to Galatians 3:8. How can it be that he is anywhere but in heaven? If "Abraham, who is the father of us all" (Romans 4:16) is not in heaven, then we (Abraham's children according to faith) will not be going immediately to heaven either.
However, Abraham is in heaven.
When people were arguing about the resurrection, Jesus said:
"But that the dead are raised, even Moses showed, in the passage about the burning bush, where he calls the Lord the God of Abraham, and the God of Isaac, and the God of Jacob. Now He is not the God of the dead but of the living; for all live to Him." (Luke 20:37-38)
Abraham's spirit cannot be in a place called Sheol, because God is called the God of the living. Abraham's body is in Sheol, but his spirit is in heaven. If both his body and spirit are in Sheol (or the grave), then God cannot be called the God of the living.
The Bible further talks about who is currently in heaven:
"But you have come to Mount Zion and to the city of the living God, the heavenly Jerusalem, and to myriads of angels, to the general assembly and church of the firstborn who are enrolled in heaven, and to God, the Judge of all, and to the spirits of the righteous made perfect, and to Jesus, the mediator of a new covenant, and to the sprinkled blood, which speaks better than the blood of Abel." (Hebrews 12:22-24)
The "spirits of the righteous made perfect" include all the Old Testament saints. And the "church of the firstborn" include all the New Testament believers.
Besides all this, during the transfiguration (which is a glimpse into heaven), Moses and Elijah appeared with Jesus.
"And behold, two men were talking with Him; and they were Moses and Elijah, who, appearing in glory, were speaking of His departure which He was about to accomplish at Jerusalem" (Luke 9:30-31). Moses and Elijah appeared in glory even before Jesus died on the cross! Did they go back to the comforting part of Sheol after the transfiguration? Or did they go back to heaven?
Of course, they went back to heaven, because they were credited with Jesus' righteousness beforehand. Therefore, Old Testament saints are already in heaven and they will receive their new bodies at the time of the Rapture before church-age believers.
The Bible says:
"But we do not want you to be uninformed, brethren, about those who are asleep, so that you will not grieve as do the rest who have no hope. For if we believe that Jesus died and rose again, even so God will bring with Him those who have fallen asleep in Jesus. For this we say to you by the word of the Lord, that we who are alive and remain until the coming of the Lord, will not precede those who have fallen asleep. For the Lord Himself will descend from heaven with a shout, with the voice of the archangel and with the trumpet of God, and the dead in Christ will rise first. Then we who are alive and remain will be caught up together with them in the clouds to meet the Lord in the air, and so we shall always be with the Lord. Therefore comfort one another with these words." (1 Thessalonians 4:13-18)
Jesus died for all (including Old Testament saints). "Those who have fallen asleep in Jesus" include Old Testament saints because the gospel was preached to them beforehand (however short) too. The Old Testament saints anticipated a Messiah. So they were credited with righteousness like Abraham.
That is why "we who are alive and remain until the coming of the Lord, will not precede those who have fallen asleep". So you're right! The Old Testament saints will receive their new bodies before us at the time of the rapture (pre-tribulation). After them, the church-age Christians who have died will receive their bodies. Finally, if we are still on earth and raptured, then we will receive our bodies.
"Behold, I tell you a mystery; we will not all sleep, but we will all be changed, in a moment, in the twinkling of an eye, at the last trumpet; for the trumpet will sound, and the dead will be raised imperishable, and we will be changed." (1 Corinthians 15:51-52)
The trumpet will begin to sound and the dead will have new bodies, each in his order. Then at the last trumpet, all the raptured people will take on glorious bodies as well.