Does any Church currently incorporate exorcism, three dunkings, and anointing in their baptism rite? Hippolytus of Rome, in relating the rites and liturgies in use in Rome in the 3rd Century AD in his work called Apostolic Tradition, describes a process of exorcism first to expel evil spirits whereby the baptismal candidate verbally renounces Satan and service to him, is anointed by Oil of Exorcism, and then passed on to another Elder for a 3 time baptism; once each for confession of each Trinity member.


  At the time determined for baptism, the bishop shall give thanks over
  some oil, which he puts in a vessel. It is called the Oil of
  Thanksgiving. He shall take some more oil and exorcise it. It is
  called the Oil of Exorcism. A deacon shall hold the Oil of Exorcism
  and stand on the left. Another deacon shall hold the Oil of
  Thanksgiving and stand on the right. When the elder takes hold of each
  of them who are to receive baptism, he shall tell each of them to
  renounce, saying, "I renounce you Satan, all your service, and all
  your works." After he has said this, he shall anoint each with the Oil
  of Exorcism, saying, "Let every evil spirit depart from you." Then,
  after these things, the bishop passes each of them on nude to the
  elder who stands at the water. They shall stand in the water naked. A
  deacon,likewise, will go down with them into the water. When each of
  them to be baptized has gone down into the water, the one baptizing
  shall lay hands on each of them, asking, "Do you believe in God the
  Father Almighty?" And the one being baptized shall answer, "I
  believe." He shall then baptize each of them once, laying his hand
  upon each of their heads. Then he shall ask, "Do you believe in Jesus
  Christ, the Son of God, who was born of the Holy Spirit and the Virgin
  Mary, who was crucified under Pontius Pilate, and died, and rose on
  the third day living from the dead, and ascended into heaven, and sat
  down at the right hand of the Father, the one coming to judge the
  living and the dead?"When each has answered, "I believe," he shall
  baptize a second time. Then he shall ask, "Do you believe in the Holy
  Spirit and the Holy Church and the resurrection of the flesh?" Then
  each being baptized shall answer, "I believe." And thus let him
  baptize the third time. Afterward, when they have come up out of the
  water, they shall be anointed by the elder with the Oil of
  Thanksgiving, saying, "I anoint you with holy oil in the name of Jesus
  Christ." Then, drying themselves, they shall dress and afterwards
  gather in the church.


After this they would anoint the newly baptized with the Oil of Thanksgiving.

Are there any traditions who currently baptize in this way (with or without the nudity)?  If not, when and why was this ritual changed?
Does any Church currently incorporate exorcism, three separate immersions, and anointing in their baptism rite (with or without the nudity)?

The short answer is probably not. Although some do more or less share similar elements.

St. Hippolytus of Rome relates how baptism in The Early Church was administered in or around 215 AD in Rome as such:


  21 At the hour in which the cock crows, they shall first pray over the water. 2When
  they come to the water, the water shall be pure and flowing, that is, the water of a spring
  or a flowing body of water. 3Then they shall take off all their clothes. 4The children shall be baptized first. All of the children who can answer for themselves, let them answer. If there are any children who cannot answer for themselves, let their parents answer for them, or someone else from their family. 5After this, the men will be baptized. Finally, the women, after they have unbound their hair, and removed their jewelry. No one shall take any foreign object with themselves down into the water.
  
  6At the time determined for baptism, the bishop shall give thanks over some oil, which he puts in a vessel. It is called the Oil of Thanksgiving. 7He shall take some more oil and exorcise it. It is called the Oil of Exorcism. 8A deacon shall hold the Oil of Exorcism and stand on the left. Another deacon shall hold the Oil of Thanksgiving and stand on the right.
  
  9When the elder takes hold of each of them who are to receive baptism, he shall tell each of them to renounce, saying, "I renounce you Satan, all your servicea, and all your works." 10After he has said this, he shall anoint each with the Oil of Exorcism, saying, "Let every evil spirit depart from you." 11Then, after these things, the bishop passes each of them on nude to the elder who stands at the water. They shall stand in the water naked. A deacon, likewise, will go down with them into the water. 12When each of them to be baptized has gone down into the water, the one baptizing shall lay hands on each of them, asking, "Do you believe in God the Father Almighty?" 13And the one being baptized shall answer, "I believe." 14He shall then baptize each of them once, laying his hand upon each of their heads. 15Then he shall ask, "Do you believe in Jesus Christ, the Son of God, who was born of the Holy Spirit and the Virgin Mary, who was crucified under Pontius Pilate, and died, and rose on the third day living from the dead, and ascended into heaven, and sat down at the right hand of the Father, the one coming to judge the living and the dead?" 16When each has answered, "I believe," he shall baptize a second time. 17Then he shall ask, "Do you believe in the Holy Spirit and the Holy Church and the resurrection of the flesh?" 18Then each being baptized shall answer, "I believe." And thus let him baptize the third time.
  
  19Afterward, when they have come up out of the water, they shall be anointed by the
  elder with the Oil of Thanksgiving, saying, "I anoint you with holy oil in the name of Jesus Christ." 20Then, drying themselves, they shall dress and afterwards gather in the church. - The Apostolic Tradition of Hippolytus of Rome


Let us remember that this was the baptismal practice as was practiced in Rome in the third century.

Although the wording used in the above formula of baptism is slightly different than what we are accustomed to hear, the Trinitarian formulation is still in tact.

The Catholic Church still permits baptism by either immersion or by pouring water over the head of the catechumen. The Early Church practiced pouring over the head during baptism for those who were elderly or seriously ill. 

The liturgy and how the sacraments are administered are never stagnate; they are administered by the Church(es) as the people of God move throughout the centuries. There is always an organic development so to speak in how the liturgy and the sacraments are administered, from century to century. 

As I have said the Catholic Church still permits baptism, either by immersion or by pouring water over the head of her catechumens. It would still be permitted to baptize an adult by immersion while using the formula now in place within the Church.

For a baptism to be valid the following must be observed according to the Catechism of the Catholic Church


  1278 The essential rite of Baptism consists in immersing the candidate in water or pouring water on his head, while pronouncing the invocation of the Most Holy Trinity: the Father, the Son, and the Holy Spirit.


I have seen priests, while pouring the water over the head of infant and pronouncing the trinitarian formula poured the water over the head in three separate pourings, one in the name of the each person of the Sacred Trinity. 

In the same way, it is conceivable, but doubtful that an immersion baptism could be administered in a similar fashion. Doubtful, because this not the General manner of administering this sacrament in our days.

As far as exorcisms and anointings are concerned, they are still administered within the Catholic Church.

The modern Rite of Baptism for One Child,for example, still incorporates exorcisms and anointings in her baptisms.

As for the administration of this sacrament while being nude that was certainly phased out because of issues of modesty were involved.