The mishna in Kinnim 1:3–4 (2–3 in some editions) describes what happens when birds from multiple offerings get mixed up together. Mishna 1:3 includes (translation from Sefaria):
חַטָּאת שֶׁנִּתְעָרְבָה בְחוֹבָה, אֵין כָּשֵׁר אֶלָּא כְמִנְיַן חַטָּאוֹת שֶׁבַּחוֹבָה. וְכֵן עוֹלָה שֶׁנִּתְעָרְבָה בְחוֹבָה, אֵין כָּשֵׁר אֶלָּא כְמִנְיַן עוֹלוֹת שֶׁבַּחוֹבָה
A sin-offering that became mixed with a [pair of] obligatory [birds], only the number of sin-offerings [contained] in the obligatory [pair] are permitted [to be sacrificed]. Similarly, an burnt-offering that became mixed with with obligatory a [pair of] obligatory [birds], only the number of burnt-offerings [contained] in the obligatory [pair] are permitted [to be sacrificed].
Mishna 1:4 includes (translation from Sefaria):
חובה שנתערבה זו בזו, אחת לזו ואחת לזו, שתיים לזו ושתיים לזו, שלוש לזו ושלוש לזו--מחצה כשר, ומחצה פסול. אחת לזו, ושתיים לזו, שלוש לזו, ועשר לזו, ומאה לזו--הממועט כשר
If obligatory [bird offerings] became mixed with each other, one [pair] from this one [woman] and one [pair] from this one [woman], two [pairs] from this one [woman] and two [pairs] from this one [woman], three [pairs] from this one [woman] and three [pairs] from this one [woman]; half are permitted [to be sacrificed] and half are invalid. [However if] this one [woman] had one [pair], and this one [woman] had two [pairs], [or] three for this one, [or] ten for this one [or] one hundred for this one, [only] the smaller number [of birds] are permitted [to be sacrificed].
We see that after a mix-up and after the allowed offerings from the mix-up are brought, a person is in one (or more) of the following situations, and they don't know which one applies to them: (1) they owe an olah without a chatat, (2) they owe a chatat without an olah, (3) they owe a pair (chatat and olah), (4) they owe nothing. Additionally, if they needed to bring multiple offerings, they do not know for which offering they may have already (partially) fulfilled their obligation.
One common way to bring an offering that may not be required is to offer it with a stipulation: if it's required, then great, but if not, then it should be considered as a voluntary offering. However, voluntary bird offerings cannot be brought as a chatat (see Kinnim 1:2).
What is the remedy for someone in this situation (or is there not one)? How (and with what stipulations) do they bring their missing offerings? If it depends on the number of pairs of birds, the number of people whose offerings were involved in the mix-up, or something else, please indicate as much in your answer.