Did only Jesus human body remain in the grave for the three days? Jesus told the thief at Calvary :


  Luke 23:43  And Jesus said to him, "Assuredly, I say to you, today you will be with Me in Paradise." 


And yet he was not resurrected until after the Sabbath. So were the Souls and Spirits of both taken to Paradise that day while their physical bodies remained ?

Using only biblical texts and/or established traditions of the Nicene church, is there a consensus or even a limited set of positions on where Jesus' spirit was?
"Using only biblical texts and/or established traditions of the Nicene
church, is there a consensus or even a limited set of positions on
where Jesus' spirit was?"

It is not clear that Nicene Council was considering this particular subject.  It seemed they were more hard pressed to establish a good date for celebrating Easter. Consult ref. below:


  http://en.wikipedia.org/wiki/First_Council_of_Nicaea The First Council of Nicaea (/naɪˈsiːə/; Greek: Νίκαια [ˈni:kaɪja];
  Turkish: Iznik) was a council of Christian bishops convened in Nicaea
  in Bithynia by the Roman Emperor Constantine I in AD 325. This first
  ecumenical council was the first effort to attain consensus in the
  church through an assembly representing all of Christendom.[5]
  
  Its main accomplishments were settlement of the Christological issue
  of the nature of the Son of God and his relationship to God the
  Father,3 the construction of the first part of the Creed of Nicaea,
  establishing uniform observance of the date of Easter,[6] and
  promulgation of early canon law.
  
  Misconceptions Biblical canon Main article: Development of the
  Christian biblical canon
  
  A number of erroneous views have been stated regarding the council's
  role in establishing the biblical canon. In fact, there is no record
  of any discussion of the biblical canon at the council at all.[66] The
  development of the biblical canon took centuries, and was nearly
  complete (with exceptions known as the Antilegomena, written texts
  whose authenticity or value is disputed) by the time the Muratorian
  fragment was written.[67]
  
  In 331 Constantine commissioned fifty Bibles for the Church of
  Constantinople, but little else is known (in fact, it is not even
  certain whether his request was for fifty copies of the entire Old and
  New Testaments, only the New Testament, or merely the Gospels), and it
  is doubtful that this request provided motivation for canon lists as
  is sometimes speculated. In Jerome's Prologue to Judith[68] he claims
  that the Book of Judith was "found by the Nicene Council to have been
  counted among the number of the Sacred Scriptures".


From John 20:17 we do know that Jesus had not ascended to the Father preceding the 3rd day after crucifixion. 


  John 20:17 NIV Jesus said, “Do not hold on to me, for I have not
  yet ascended to the Father. Go instead to my brothers and tell them,
  ‘I am ascending to my Father and your Father, to my God and your
  God.’ ”


Luke 23:43 lets us know that wherever else he might have gone on day one, he was at the very least in paradise.

Benson [Benson Commentary] has an excellent commentary on the subject and you may find it of use.  With this partial quote from the text:


  "...That he was going, not only to αδης, the invisible world, but to
  that part of it termed paradise (my markup). His human soul was
  removing to the place of separate souls; not to the place of the
  damned, but to the place of the blessed...."