זֹ֣את בְּרִיתִ֞י אֲשֶׁ֣ר תִּשְׁמְר֗וּ … הִמּ֥וֹל לָכֶ֖ם כּל־זָכָֽר׃
“This is my covenant that you shall observe, circumcise for you all males.” (Breishit 17:10)
Hopefully this is only a theoretical question. Can a convert, who was already circumcised as an infant, perform the requisite hatafat dam brit (the taking a drop of blood through a pin prick from the circumcision site) on himself?
No. This person converting is not yet a Jew, and therefore hatafah would not be valid if performed by him.
Regarding milah for a Jewish baby, the Shulchan Arukh (YD 264:1) rules that the mohel has to be Jewish, but if he were not Jewish the baby does not need a second milah. Some commentators understand that to mean that bedieved a non-Jew can be a mohel for a Jewish baby (See Taz there), but many others understand that it means that the milah was not good, and now there is nothing that can be done to make it better. It seems to me based on the Gemara (Avodah Zarah 27a) the latter reading is definitely the more straightforward one.
That is for milah for a Jewish baby. When it comes to conversion, we find that the Gemara discusses a case of a person who said that they were converted on their own, without anyone else presence, and its only concern is whether he is to be believed and the need for a beit din, but not the need for the circumcision to be performed by a Jew. This would suggest that a non-Jew might, ironically, be able to perform the act of circumcision for a conversion, although the widespread consensus of the poskim is that that is not the case, even if it is being rabbinically supervised, and that this is even more clearly invalid than the case of a circumcision of a Jewish baby (see Achiezer 3:27).
In short, a Jewish person must perform the hatafat dam brit of a man who is converting.