From Maimonides' Mishneh Torah (Repentance Chapter 1:1) it seems Teshuva - returning/repentance (at least nowadays*) is an essential ingredient of atonement for all categories of sin:
...[the sacrifices] do not atone for them until they repent and confess verbally, as it says: he shall confess the matter in which he sinned (Leviticus 5:5). ...their death or lashing does not atone for them until they repent and confess verbally. And also someone who has injured his fellow or damaged his property, even though he has repaid what he owes him, he has not atoned until he confesses and turns away from similar actions forever, as it says: From any of the sins of man (Numbers 5:6).
From the fact that without repentance, even suffering and/or death do not atone, it seems fairly implicit that there is no time limit, at least in this world**.
Similarly, the implication of 1:4 is that time and suffering only function toward atonement when teshuva is also involved:
Even though teshuvah atones for all, and the day of Yom Kippur itself atones - there are some sins that can be atoned for in their time, and some sins which are only atoned after time has passed. What case is that? If a person violated a positive commandment for which the punishment is not karet and did teshuvah - before he can even move he is forgiven, and regarding such people it is said: Return, backsliding children; I will heal your backslidings (Jeremiah 3:22). If a person violated a negative commandment for which the punishment is neither karet nor capital punishment and did teshuvah – the teshuvah suspends it and Yom Kippur effects atonement; and regarding such people it is said: For that day will atone for you (Leviticus 16:30). If a person violated [a commandment for which the punishment is] karet or capital punishment and did teshuvah – the teshuvah and Yom Kippur suspend it and the suffering that falls upon him effects atonement. And no matter how much time passes, he does not receive full atonement until suffering falls upon him; and regarding such people it is said: Then will I visit their transgression with the rod, and their iniquity with strokes (Psalms 89:33). What cases are we discussing? Cases in which he did not disgrace the name [of G-d] at the moment of violation. But someone who disgraces the name [of G-d], even if he does teshuvah, and Yom Kippur passes and he remains in his teshuvah, and suffering befalls him – he does not receive full atonement until he dies; the teshuvah, Yom Kippur, and the suffering all suspend and death atones, as it says: And the Lord of Hosts revealed Himself in my ears: Surely this iniquity shall not be atoned until you die (Isaiah 22:14).
Relatedly, we find a debate amongst the Tannaim and Amoraim regarding the dependence of the final communal salvation upon repentance (Sanhedrin 97b):
:אמר רב: כלו כל הקיצין ואין הדבר תלוי אלא בתשובה ומעשים טובים. ושמואל אמר
דיו לאבל שיעמוד באבלו. כתנאי...
Rav said: all the deadlines have passed and the thing is only dependent on repentance and good deeds. And Samuel said: It is enough for the mourner to stand in his mourning. This is also disputed by the Tannaim...***
(See there the full discussion.)
*Interestingly, however, in the time of the Temple service, atonement (at least on the communal level) did not always require teshuva:
1:2 ...But if repentance wasn't done, the goat only atones for the light ones. What are "light ones" and what are "heavy ones"? The heavy ones are sins that have the obligation of death by court, or excision. And false or vain oaths, even though they do not have excision, are of the heavy ones. And the rest of the negative commandments, and the positive commandments which do not have excision - those are the light ones.
However, considering that the scapegoat is a communal offering, it seems well within reason to argue that Maimonides is commenting here only regarding communal atonement, not of individual atonement, which may well always require some form of actual repentance.
**There is a tradition that postmortem suffering has a general time-limit of one year (see e.g. Eduyoth 2:10; See also Rosh Hashanah 17a). To what level these types of ideas can be understood simplistically/literally is controversial at best. But the suggestions does seem to be one of finite suffering even absent actual repentance.
***See however the subsequent passage where indeed it seems that even Samuel's position is that repentance is necessary, but if the final deadline is reached, the level of threat becomes one that precludes real free will, i.e. the teshuva is forced (perhaps akin to the midrashic concept of the acceptance of the Torah - kafa aleihem har k'gigit). See also alternate versions of the debate. In any case, it could certainly also be argued that this view of non-dependence on free-willed teshuva might also only be at the communal level of redemption, not with regard to individual atonement.