Reading down some verses, the actual content of James' letter is given:
For it seemed good to the Holy Spirit and to us, no greater burden be added to you except these necessary things: to keep away from things offered to idols, and blood, and things strangled, and fornication. By keeping yourselves from such things you will do well. Farewell.
-- Acts 15:28-29
So, it becomes clear that the "pollutions of idols" of Acts 20:22 that concerned James, was εἰδωλόθυτος (Strong's G1494 - eidōlothytos, i.e sacrificed to idols, the flesh left over from the heathen sacrifices)
ἀλισγημάτων, given as "pollutions" in Acts 22:20, is the plural of ἀλίσγημα (Strong's G234 - alisgēma), which is the only occurrence in the NT. However, there are two instances in the LXX of a related verb ἀλισγέω, which means: to pollute, destroy the purity of, destroy the sanctity of:
And Daniel purposed in his heart, that he would not defile himself ἀλισγηθῇ with the king's table, nor with the wine of his drink: and he intreated the chief of the eunuchs that he might not defile himself.
-- Daniel 1:8 (Brenton LXX)
The life of him that dependeth on another man's table is not to be counted for a life; for he polluteth himself αλισγησει ψυχην αυτου with other men's meat: but a wise man well nurtured will beware thereof.
-- Sirach 40:29 (Brenton LXX)
It is interesting to note that Brenton chose "defile himself" for the verb in Daniel 1:8, but "polluteth himself" in Sirach 40:29. It is unlikely this was unintentional. Daniel was a son of Israel, and one of his great concerns was ritual purity. The wisdom being shared in Sirach, on the other hand, is generic (no mention of idols, only "other men's meat"), and is applicable advice for all men, even those who have no concern with ritual purity.
James is writing on behalf of the Apostles to advise the "Gentiles" of the essentials of the Law that would be pertinent to their lives, and even though he mentions idols, his concern is not with ritual purity, but their good health. The final salutation in the letter is ἔρρωσθε (Strong's G4517 - rhōnnymi: "farewell"), which could just as easily been given as "Good health!"
James is apologetic in his letter:
Forasmuch as we have heard, that certain which went out from us have troubled you with words, subverting your souls, saying, Ye must be circumcised, and keep the law: to whom we gave no such commandment:
-- Acts 15:24 (KJV)
So James' mind clearly could not have been fixed on the ritual aspects of the Law, but rather on those aspects that were relevant to lives of the Gentile believers, i.e. their good health.
ἀλισγημάτων could be given in Acts 22:20 as either "pollutions" or "defilements", but since James' target audience is Gentile, and his intent is not religious in nature, one might just as well follow Brenton's lead and give it as "pollutions".