What is the biblical and patristic basis for the belief that the saints go to Heaven instead of Sheol after the resurrection and ascension of Jesus?
To start with we have the words of Jesus to the Good Thief on the cross.
43 And Jesus said unto him, Verily I say unto thee, Today shalt thou be with me in paradise. - Luke 23:43
Then we have Stephen’s vision of heaven just before he was martyred.
55 But Stephen, full of the Holy Spirit, looked up to heaven and saw the glory of God, and Jesus standing at the right hand of God. 56 “Look,” he said, “I see heaven open and the Son of Man standing at the right hand of God.”
57 At this they covered their ears and, yelling at the top of their voices, they all rushed at him, 58 dragged him out of the city and began to stone him. Meanwhile, the witnesses laid their coats at the feet of a young man named Saul.
59 While they were stoning him, Stephen prayed, “Lord Jesus, receive my spirit.” 60 Then he fell on his knees and cried out, “Lord, do not hold this sin against them.” When he had said this, he fell asleep. - Acts 7:55-60
Then there is the great multitude in white robes in heaven from the Book of Revelation.
The Great Multitude in White Robes
9 After this I looked, and there before me was a great multitude that no one could count, from every nation, tribe, people and language, standing before the throne and before the Lamb. They were wearing white robes and were holding palm branches in their hands. - Revelation 7:9
The Book of Revelation shows the saints worshipping God, singing hymns, playing instruments, making requests to Christ to avenge their martyrdom, and offering prayers for the saints on earth (Rev. 4:10, 5:8, 6:9-11).
As far as a patristic basis is concerned, it took centuries, for the Early Church to grasp the notion that the saints in heaven could be anything else than martyrs. For the Early Church only Martyrs who shed their blood for their faith were recognized as saints in heaven.
Perpetua only saw martyrs in heaven, according to Tertullian, De anima 55,4. This passage has perplexed scholars, since Tertullian seems to be referring to Saturus’s vision, not Perpetua’s (Passio Perpetuae et Felicitatis 13,8). Additionally, Tertullian’s citation is part of his larger argument against the Valentinians, in which he makes the peculiar claim that the souls of the dead are “below” (inferi) with the exception of the martyrs who are in Paradise. I contend that Tertullian’s claim has been misunderstood in the last few decades of scholarship because of a failure to contextualize his remark within his rhetorical strategy. Disentangling Tertullian’s convictions from his rhetoric is notoriously difficult, and yet by reading Tertullian as fully immersed in the tactics from the Second Sophistic Movement recent scholars have made great advances in our understanding of this North African Christian writer. Several of Tertullian’s other works provide counter-evidence to the idea that only martyrs go to heaven: specifically, Tertullian further defines “heaven,” its location, and its occupants; additionally, Tertullian clarifies who is a “martyr” in his wider oeuvre. When Tertullian’s own teachings on the afterlife are retrieved, then one can re-read De anima to see how Tertullian has cloaked these with rhetorical devices meant to refute the Valentinian notion of the soul’s ascent through multiple heavens. This idea that Tertullian believed only martyrs gain immediate access to heaven—which has often been repeated in the most recent century’s secondary literature—is itself a misunderstanding of earlier modern scholarship. Tertullian on the Afterlife: “Only Martyrs are in Heaven” and Other Misunderstandings
To a certain degree, one could possibly understand this. The physical evidence from witnesses was enough proof of their holiness. Here I am thinking of Polycarp’s martyrdom in 155.
This explains to some degree also why there is a lack of Early Church writings on the saints as we understand the term in modern usage. For them a saint was a martyr!
The development of the Feast of All Saints bares this out. The original title being the feast of the dedicatio Sanctae Mariae ad Martyres (Blessed Virgin Mary and all the martyrs). Notice that the Church’s first title of a non-martyr was yet united to the Mother of Jesus.
From the 4th century, there existed in certain places and at sporadic intervals a feast day to commemorate all Christian martyrs. It was held on 13 May in Edessa, the Sunday after Pentecost in Antioch, and the Friday after Easter by the Syrians. During the 5th century, St. Maximus of Turin preached annually on the Sunday after Pentecost in honor of all martyrs in what is today northern Italy. The Comes of Würzburg, the earliest existing ecclesiastical reading list, dating to the late 6th or early 7th century in what is today Germany, lists this the Sunday after Pentecost as dominica in natale sanctorum ("Sunday of the Nativity of the Saints"). By this time, the commemoration had expanded to include all saints, martyred or not.
On 13 May 609 or 610, Pope Boniface IV consecrated the Pantheon at Rome to the Blessed Virgin Mary and all the martyrs, ordering an anniversary; the feast of dedicatio Sanctae Mariae ad Martyres has been celebrated at Rome ever since. It is suggested 13 May was chosen by the Pope and earlier by Christians in Edessa because it was the date of the Roman pagan festival of Lemuria, in which malevolent and restless spirits of the dead were propitiated. Some liturgiologists base the idea that Lemuria was the origin of All Saints on their identical dates and their similar theme of "all the dead".
Pope Gregory III (731–741) dedicated an oratory in Old St. Peter's Basilica to the relics "of the holy apostles and of all saints, martyrs and confessors, of all the just made perfect who are at rest throughout the world". Some sources say Gregory III dedicated the oratory on 1 November, and this is why the date became All Saints' Day. Other sources say Gregory III held a synod to condemn iconoclasm on 1 November 731, but dedicated the All Saints oratory on Palm Sunday, 12 April 732. - All Saints' Day
In any case this is an extremely interesting topic and hope others will come forth with better posts than this one.