Your question is unanswerable without making some assumptions - so here we go.
Adoption is not a conversion process. The child's religious status does not change with the legal paperwork. A Jewish child adopted by a Jewish family is still Jewish. A non-Jewish child adopted by a Jewish family is still not Jewish. The intervening adoption is not a concern here. Assuming it was done correctly for legal purposes, it is not at issue. The status of the adoptive parents is not a factor to answer the underlying question. This is really a question of status - Jewish or not.
First possibility: The child was born of a Jewish mother, and was adopted by a Jewish family. In that instance, the child would be Jewish at birth, and still Jewish upon adoption.
However, that birth mother must be Jewish AT THE TIME OF the birth; otherwise, she is not Jewish, and therefore the child born is not Jewish, absent a conversion.The child takes the status of the mother at the moment of birth.
A later status change of the mother does NOT retroactively affect the child born earlier, and a conversion of the child would be required for that child to be Jewish.
Second possibility: The child was born of a non-Jewish mother, with a non-Jewish father (i.e., is absolutely not Jewish at all), and was adopted by a Jewish family. Clearly here, too, for the child to be considered Jewish, there must be a conversion.
Third possibility: The child was born of a non-Jewish mother, with a Jewish father (i.e., that child is not Jewish according to Halachah, but could conditionally be considered Jewish in the Reform context, IF and ONLY IF the child is: given a Jewish education, raised exclusively as a Jew, affirms Jewish status by publicly (and privately) engaging exclusively in Jewish rituals and practices (including, as appropriate, Brit Milah/Brit Bat & Naming, B'nai Mitzvah, Confirmation, Jewish wedding, observing holidays and rituals, etc.).
In the first case, the child is Jewish by Halachah and tradition.
However, the way you pose the question, apparently the status of the mother is unknown or questionable.
As a way to resolve any questions that exist in that situation, I would advise that the child be taken to the Mikveh in the presence of a Beit Din (a rabbinic court of three) for immersion.
In that situation, if the birth mother was Jewish, this effects no change in status, except to settle any doubt (Safek), and to provide a record for future use if the question arises. If the birth mother was not Jewish, it would serve as a conversion ritual, and also settle the status of the child for the future.
In the second case, this child is not Jewish by any definition. The child would require the full conversion process: Brit Milah if a boy, and immersion in the Mikveh before a Beit Din to effect a conversion. This would be what I would recommend to the adoptive parents, in any case.
The third case is a little more difficult to resolve because the answer differs between and within Jewish movements. For the Orthodox and Conservative responses, see what my colleagues have written.
Unfortunately, it is true that there is not complete agreement among all within the Reform movement as to what to do in this situation, so I am giving you my own answer: it fits within the ambit of the Reform movement's view, but not all Reform rabbis (or Reconstructionist or Renewal or other groups' rabbis, for that matter) would follow the approach I suggest to you in my response. This is a somewhat gray area. I tend to lean towards honoring, and therefore following Halachah and tradition as much as possible.
My approach is as follows (with my parenthetical comments inserted): the Reform movement has stated in Responsa that a child of a Jewish parent -either of the biological parents, that is raised in a Jewish (Jewish only - not a two-faith family) home, given a Jewish education (and no other religious instruction), (participates in the Jewish community), and that participates and engages in the public (and private) rituals, practices, and activities that affirm their Jewish status can be considered Jewish.
In this hypothetical situation you present, a child born of a non-Jewish biological mother and a Jewish biological father, adopted by a Jewish couple, and raised as a Jew, including full participation in Jewish rituals and practices, engaging in a formal Jewish education, and affirming their Jewish connection and affiliation by public acts, COULD - and probably would - be considered Jewish in the Reform (and perhaps other progressive groups in the Jewish world) movement, but WOULD NOT be so considered by Conservative or Orthodox groups, absent a formal conversion.
Consequently, I would advise this person to undertake a process of study and working with a rabbi to effect a resolution to any doubt that exists. By following a course of study, undergoing Brit Milah (ritual circumcision) or Hatafat Dam Brit (drawing a bead of blood if circumcised) for a male, appearing before a Beit Din, and immersing in a Mikvah, it would remove the possibility of any Safek, and for those groups in the Jewish world that do not accept patrilineal descent (taking Jewish status from the father), this would have the effect of a conversion ritual.
Not to paint too rosy a picture (it coulnd't be that easy!): conversions by Reform, Reconstructionist, Renewal, and even Conservative rabbis are not necessarily accepted by the Orthodox (or as some would term them, the ultra-Orthodox, or Haredi) rabbinate. Depending on the purpose of the conversion, I would urge you to ask questions about the acceptance of a Teudah (certificate) from the particular rabbi and the Beit Din, for the use you wish to make of it.
If you wish to be accepted in the very Orthodox world, for example, a conversion with a Reform rabbi is not likely to be good enough. This is not about Halachah (although the arguments are often phrased in those terms); it is about politics and power and defining 'our group' and being "right" - and there is plenty of blame to go around on this issue for all.
I urge you to be fully informed and to work towards fulfilling what is necessary for your particular situation. Ask a lot of questions. There is no 'one size fits all' solution possible here.
Rabbi Joe Blair