The answer is that it depends how far the kohen has progressed in the second aliya. If he has already started the beracha, he continues it, but if he has not yet started it, then we switch to the levi. This halacha appears in the Shulchan Aruch,[1] both regarding your case as well as regarding the parallel case, when it is not realized that a kohen is present and a yisrael is mistakenly called for the first aliya.
The reason we switch the olim when we can is that giving a first aliya to a non-kohen[2] and giving a second aliya to the kohen who got the first one (in order to protect the reputation of the kohen[3]) are both exceptional scenarios that are avoided when possible. We will add that replacing the person who is poised to begin does not hurt him in any way. A yisrael who is passed up for a kohen never had claims to the first aliya to begin with; in addition, so as to not insult him, we keep him at the bima until we are able to give him the third aliya, which is the first one available to a yisrael.[4] In the case of a kohen who has already had an aliya, he is simply being held back from getting a second one. Giving a kohen two aliyot is done only when there is no alternative, and the fact that he does not get the second aliya he was preparing for cannot be construed as our questioning his status as a kohen.
In the case in which the “wrong person” has already started his second aliya, the halacha is that he completes the aliya. We cannot stop him in the middle, because he would then have made a beracha l’vatala.[5] Halachic decision-making often comes down to weighing and choosing between two less than ideal situations. In our case, causing a beracha l’vatala is a serious problem, whereas the concern that people will think that a yisrael was given the first aliya instead of a kohen because there is something wrong with the kohen’s lineage is not severe. After all, people can understand that the reason the kohen was not called was that he was temporarily unavailable or that he was accidentally overlooked.[6] In such a case, we do not call up a kohen afterward for the second aliya, because taking that step would make the second oleh look like a non-kohen, as he would be getting an aliya after a yisrael already did, which a kohen is not supposed to get.[7]
In your case, if the kohen was not replaced after starting his second aliya, there is actually a bigger “sacrifice” that is being made to avoid the beracha l’vatala. Even though the kohen did not need to get the exceptional second aliya, we still count his second aliya toward the number of required aliyot and suffice with two people getting aliyot instead of the normal minimum of three.
A noteworthy point is the question of what is considered having “started the aliya.” The Shulchan Aruch[8] rules that the correct person can replace the one who should not be receiving the aliya even after the latter recited Barchu. Barchu is not considered a “point of no return,” because when one commands the tzibbur to bless HaShem (which is Barchu’s role), and they do so (by answering “Baruch HaShem HaMevorach…”), that has value independent of the aliya.[9] According to most authorities, the correct person who takes over repeats Barchu before his aliya.[10] Although some maintain that this is unnecessary,[11] it makes sense to follow the majority opinion, because reciting an additional Barchu is not a significant problem even if it is essentially unnecessary.[12]
Regarding the opening beracha itself, the Magen Avraham[13] rules that until one has said HaShem’s Name (the third word), it is permitted and correct to stop the beracha in order to transfer the aliya to the correct person. Once the oleh has said HaShem’s Name, however, he must continue the beracha and therefore get the entire aliya. Although there is generally a remedy to end a beracha at that point by turning the beracha into the pasuk of “Baruch ata HaShem lamdeini chukecha,”[14] this remedy is not perfect and is not deemed justified in this case.
There is an opinion that the above is true only when the correct person just entered, whereas if he was there to begin with but was simply not noticed, we would stop the beracha after three words and add “lamdeini chukecha.”[15] However, the consensus of poskim is to not make such a distinction.