Most Christian sects, rather than giving an explicit teaching on this, would tend to say that certain views are unacceptable, and others are acceptable, under their broader theological perspective.
Since I'm a Catholic, I'm arranging this according to views acceptable under Catholic doctrine and views unacceptable under Catholic doctrine. Non-Catholic sects may hold some of the views unacceptable under Catholic doctrine as acceptable, and vice versa, some sects may hold some of the views acceptable under Catholic doctrine as unacceptable.
This list will probably not be exhaustive, although I will try to cover the most common views. Please do let me know in a comment if you believe I have missed a common view.
Views Acceptable Under Catholic Doctrine
All Aborted Babies are Saved via Martyrdom
This view typically views aborted babies as "baptized by blood," that is, they are a kind of martyr. Because of their martyrdom, they are saved without having to be baptized by water. The biblical type for this road to salvation are the Holy Innocents, the children who were killed by Herod in Bethlehem. The Church holds that these babies, though they were unbaptized and had no capacity to ask forgiveness of God, were nevertheless cleansed of original sin because they were killed in hatred of the faith (Herod was trying to kill Christ by killing them). If one holds that all abortions are done in a kind of hatred of the faith (a debatable point), one could hold that all aborted babies are saved as martyrs.
Some Aborted Babies are Saved via Invincible Ignorance
The Church holds that people can be saved via invincible ignorance, which is an ignorance of one's own sinfulness and need to repent due to no fault of one's own. Owning to their stage of development, this kind of ignorance could be easily imputed to all aborted babies. Possibly, it applies to some and not all, but it is difficult to see how it could apply to only some and not others. Therefore, "some" in this view could include all, and for many people who hold this view probably does include all.
Aborted Babies are Generally Damned
Because the baptisms of infants are valid in Catholicism owing to the intention of their baptized parents, one could generally hold that miscarried children of Christians go to heaven via the baptism of desire. It follows, then, that the children of people who are aborted generally go to hell (with perhaps some exceptions), since there is evidently no baptism of desire on the part of the parents for their child. An exception to this rule could be the case of a Christian father who desires baptism for his unborn child, but who has no say in the decision to abort. Possibly, those babies whose parents who are invincibly ignorant and abort are saved via their own invincible ignorance. This is likely a minority view in the Church today, but I would be unsurprised if this kind of thinking (that before the age of reason the disposition of a parent is important to the salvation of their unbaptized child) were more prevalent in bygone ages.
Aborted Babies go to Limbo
There is a longstanding tradition that babies who die before baptism and before the age of reason don't go to heaven or hell, but to an inbetween place where they receive neither the pains of hell nor the Beatific Vision, generally called limbo. The sensus fidelium seems to be moving today towards the view that this is erroneous, and that these babies tend to be saved via one of the above mentioned methods.
Views Unacceptable Under Catholic Doctrine
Everyone goes to heaven, at least eventually if not right away upon death. This applies to the unborn as well.
All Aborted Babies are Damned
Usually this kind of view stems from the view that no one can be saved who is not baptized. This view may exist in some extremist high-church Protestant or low-church Baptist sects. Since aborted babies (and miscarriages, still-borns, etc) are not baptized, they go to hell, too. Although Catholics believe that baptism is generally necessary for salvation, as mentioned above, there are extraordinary means to salvation as well, so a Catholic must reject this view.
All Aborted Babies are Saved as Innocent
A denial of the effects or doctrine of original sin, this view would hold that babies who die unbaptized must go to heaven because they are sinless. That is, they are not guilty of any personal sins. Therefore, they would go to heaven.
Aborted Souls are Annihilated
There is a strain of thought in some Christian groups that the souls of the damned are annihilated. Rather than suffer the pains of hell, the soul ceases to exist. I'm not sure if any do, but I could see some Christians holding the view that before a certain age, the souls of those who die unbaptized are annihilated and cease to exist. This is trivially true in the case of those who believe in Annihilationism and also hold that unbaptized babies who die are damned, but I am thinking here that perhaps one believes that the wicked are damned and suffer the pains of hell, while God mercifully annihilates the souls of those who die unbaptized, but personally innocent (that is, free from any personal commission of sin, but not from the effects of original sin).