There is a negative commandment forbidding believing negative things about Jews (whether it is permissible to believe about non Jews is a point of contention, and for the purposes of this question let's assume it is permitted).
Mechilta Mishpatim Parsha 20/ 23:1
לא תשא שמע שוא הרי זה אזהרה למקבל לשה"ר
Rambam Hilchos Sanhedrin Chapter 21 Halacha 7
(שמות כג-א) "לֹא תִשָּׂא שֵׁמַע שָׁוְא". וּבִכְלַל לָאו זֶה אַזְהָרָה לִמְקַבֵּל לָשׁוֹן הָרַע
Since it is forbidden to believe negative things about another, I have two questions (which are two sides of the same question)
1) If someone believed loshon hara about a non Jew (assuming that is muttar) who later converts, are they required to stop believing it?
Say someone said the non Jew has a bad temper, which isn't something they necessarily were able to be fix before converting (which is possible since it doesn't halt the conversion process, nobody is perfect). If the listener assumes the new convert still has a bad temper, are they transgressing this mitzvah (besides not judging favorably, which might apply in this case or just be a middah tovah)?
2) If a non Jew believed loshon hara about a Jew and then converts, are they obligated to stop believing it?
Here the new convert has no reason to believe the Jew has done teshuvah (unless the Jew is a tzaddik, but I'm not referring to such a case). If they continue believing the negative information about the Jew, are they transgressing this mitzvah?