If a Cohen or Levi adopts a non-Jew, upon their conversion are they a Cohen or Levi as well or are they a Yisroel?
A convert is ben/bat Avraham (v'Sarah) -- not ben/bat the adoptive parents and also not ben/bat the birth parents, halachically speaking. If he doesn't even get his adoptive father's name, it seems unlikely that he would get his tribe.
They are neither a Cohen nor a Levi nor a Yisrael. They are a Ger. Gerim are a different status as evidenced by the list of different ancestral statuses in the Mishna in Kiddushin 4:1, and by certain rules pertaining to them such as their ability to marry a mamzer, and their inability to marry a Cohen (Rambam Issurei Biah 15:7 and 18:3).
They would become ישראלים, that is, neither a kohen nor a levi.
Converts to Judaism cannot merge into a family: all are created equal.