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Answer: It depends. The Shulchan Aruch (Orach Chayim 135:6-7) generally discusses your two cases, when the mistake was discovered after the oleh began the opening beracha. Both when a yisrael began the beracha for the first aliya before the kohen entered the shul (ibid. 6) and when a kohen began his second aliya when it turned out a levi was present, the mistaken oleh finishes the aliya. The clear implication is that when they had not started, we switch to the correct person even though the wrong one was called up.
The logic of switching is two-fold in the respective cases. Giving a second aliya is an exceptional act (needed to protect the reputation of the kohen –Shulchan Aruch ibid. 8), as is giving a first aliya to a non-kohen (Shulchan Aruch ibid. 4). Therefore, we do this only when there is an important reason. We are not depriving the person who is being asked to step aside of something he deserves: The yisrael never had claims to the first aliya, and we keep him at the bima until we can give him the third aliya (Shulchan Aruch, ibid. 6).The kohen already had his aliya, he is just being held back from an unusual aliya (and according to some, a b’di’eved one – see discussion in Maharam Shick, OC 61), and the levi getting the aliya after him raises no questions about his standing as a kohen.
Bemare Habazak - Rabbis Questions (654)
Rabbi Daniel Mann
435 - Davening Outside with Gloves
436 - Finding Out Late about the Presence of a Kohen or Levi
437 - Eat Meat on Shabbat when it Bothers his Wife?
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A not simple point becomes evident from the case of the kohen not being replaced after starting his second aliya. That is that even in the case that he really should not have received this exceptional second aliya, that second aliya still counts toward the number of required aliyot.
What is considered having started the aliya is noteworthy. The Shulchan Aruch (ibid. 6) rules that Barchu is not considered the beginning, so that the correct person can switch with him after Barchu. That is because commanding the tzibbur to bless Hashem (which is Barchu’s role) and their doing so ("Baruch Hashem Hamevorach…") has an independent value (Mishna Berura 135:21). According to most, the correct person who takes over repeats Barchu before his aliya (ibid.). Although some say this is unnecessary (Aruch Hashulchan, OC 135:15), it is not a problem to do an arguably extra Barchu (Kaf Hachayim, OC 135:39).
One point that is not agreed upon is whether the first aliya of a non-kohen was valid when the kohen was present and just was not noticed, as the Shulchan Aruch (OC 135:6) addresses the case when he had not yet arrived. The Pri Chadash (135:6) infers from the gemara (Gittin 59b) that passing on the takana to have the kohen go first renders the berachot and the aliya invalid even b’di’eved. The Magen Avraham 135:11 disagrees, reasoning that since regarding the kohen who took the levi’s aliya, the Shulchan Aruch (ibid. 7) says that the aliya is valid even if the levi was present, the same is true of the skipped kohen. One can argue on behalf of the Pri Chadash that the takana to give the first aliya only to the sanctified kohen is stronger than the halacha that a levi gets the second aliya before allowing the kohen to get another one. However, the Noda B’yehuda cites an interesting proof against the Pri Chadash, and this is what is accepted (Mishna Berura 135:20).

Bemare Habazak - Rabbis Questions Ask the Rabbi: Omitting Tachanun in Selichot in the Presence of Simcha
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