1. An onen  a person who is obliged to mourn, i.e. a close relative of a deceased, after death but before burial.
2. An avel is a person is obliged to mourn after burial, i.e. “the closing of the stone” or the covering of the grave.
3. Qaddish is that prayer that concludes the formal prayer servuce and the practice evolved that the final Qaddish was reserved for the orphan, i.e. one whose parent [i.e. not including offspring or sibling] had died.  The saying of Qaddish has evolved over time into a universally accepted custom; this universality renders its recitation as a virtual rabbinic law called minhag Yisrael, a universal Jewish practice.
4. Post-Talmudic folklore maintains that the Qaddish as we have it and the way it is recited possesses the power to raise the dead from Hell, angels do not understand Aramaic and will be unable to critique the soul of the deceased if the prayer were recited in Hebrew. According to normative rather than folkloristic Jewish doctrine, the saying of Qaddish is a matter of sanctifying God’s name and it is not a mantic formula which convinces God to be easy on an otherwise unworthy person. See Deuteronomy 32:4, which upon reflection would render the folk religion doctrine unorthodox. The Aramaic text more likely testifies to the fact that at the time Jews composed the Qaddish, they seem to have been speaking Aramaic.
5. The  prayer leader—and not the mourner—says the Qaddish when the practice to say it arose. [Soferim 21:5]  R. Ovadia Yosef [Yehavveh Daat 5:59]  reports that the Qaddish recitation the purpose of which is to raise the soul of the dead [a] first appears in medieval Ashkenaz and [b] its source is in the people’s practice, and the people are “prophetic offspring.” A brilliantly subtle writer with his own encoded vocabulary, R. Yosef is both patronizing his simple reader for whom folk piety is important and, at the same time,  informing his sophisticated reader who is being reminded that [a] Qaddish is a popular practice, not a mandated law, [b]  that should be honored and observed nevertheless [supra., 2:71]. Here, R. Yosef says, when dealing with folkways of which approves that being “prophetic offspring” carries legal valence. In “official religion” Orthodox Judaism,” this doctrine does not carry normative valence.  His idiom, appearing elsewhere,  that a practice’s roots are found on  “holy mountains”  serves the same purpose, of legitimating a popular practice by apodictically declaring it to be so, without a legal argument. The practices so defined have their sources of popular usage and not formal Halakhic norm.

1. The onen is not permitted to pray.  Therefore the onen does not say Qaddish.
2. There is however a positive commandment to bury the deceased as soon as possible [Deuteronomy 21:23, Sanhedrin 46b]
3. Delaying burial for scientific research on the deceased’s remains has never been normative Halakhic behavior.  Usually, delay of burial for immediate medical life-saving information may be required.
4. While saying the Qaddish for a relative is a relatively late addition to normative Jewish practice, it has become a wide spread and popular practice.
5. The requirement of prompt burial has biblical roots, where the “mourner’s” Qaddish is grounded in folkways, folklore, and folk religion.
6. Nevertheless, if a rabbi believes that a questioner is unable, unwilling, or unready to hear the correct answer, the rabbi at his discretion has  the pastoral discretionary authority to withhold information from the questioner lest that person sin intentionally rather than in error.  Betsa 30a reports this discretionary principle regarding the by law clapping and dancing on the Sabbath and Holy  done in modern times by otherwise Orthodox Jews. The Tosafists [ad. loc.] tried, apparently successfully, to abolish this restriction.  Very few Orthodox Jews, or for that matter, rabbis, are aware of this discretionary principle or the fact that dancing and clapping are forbidden on the Sabbath and Festivals, including Simchat Torah.

1. When partially committed Jews wish to “plug in”  to Jewish law, they should be accepted as they are, and encouraged to grow, probe, and experience Judaism, as fully as they are able.
2. While explaining that the delay in burial is problematic, their desire to say Qaddish is nevertheless commendable.
3. Jewish law maintains that mourning  begins when either burial is compete or the survivors turn from he deceased.
4. Therefore, once the deceased’s remains are put in the hands of the medical researchers’ possession, whether or not that act is proper, the survivors pass from aninut, being an onen, for whom prayers, including Qaddish, are inappropriate, to avelut, being an avel, a mourner, for whom prayer is required and Qaddish is appropriate.

There are any number of reasons why someone might not be buried in the presence of their children. In pre-modern times, those included frozen ground, that couldn't be broken until the spring thaw (without jackhammers and back hoes). Or people's remains might be sent to a distant town or to the Land of Israel for burial, a journey that once took a week or more. In our day, those reasons include the cases of parents who demand to be cremated or to donate their remains to research. While those modern choices may not conform to Jewish tradition, that need not affect the mourning by their children. According to Jewish norms, when mourners will not be present at the burial of their loved ones, they should begin shiva "when they turn their faces away" from the funeral cortege. In other words, when you have done everything that you can possibly to aid in the burial of your loved one, and you "turn your face" from the process of disposing of the body and bid your loved one farewell, then shiva should begin, and the mourner can say kaddish. In the case described, I would say that if your friend feels duty bound to honor his mother's wishes, then once he consigns her body to the university, he has done what he can to dispose of her remains honorably, and should begin shiva and say kaddish.

 

As I understand your question, the concern is that kaddish either cannot be recited or will be delayed because of the donation of the body for science.

From a Reform point of view the donation of one's body for scientific study “is most certainly an instance of pikuach nefesh (saving a life)."  In contrast to those who might consider the donation of a body for medical education as something less than pikuach hanefesh, we hold that "if autopsy is an essential feature of medical education, it makes little sense to delare that we aprove of the saving of a life but not of the means by which medical professionals are trained to accomplish that goal... Our position presumes that the remains will be treated with the respect due to the human body (kevod hamet).” [Jewish Living: A Guide to Contemporary Reform Practice, Mark Washofsky, UAHC Press, 2001,, pg 189]

Based on this understanding there would be no reason to assume that the recitation of kaddish would be affected. Beginning with the time of death, or following a memorial service, the mourners would recite kaddish just as they would following a standard burial.