WARSAW—The Catholic Church in Poland has seen an uptick in accusations from people claiming to have been sexually abused by clergy as children, after a YouTube video sparked public anger at an institution that is at the political and social heart of this culturally conservative country.

The increase appears to have been prompted by the two-hour-long YouTube documentary about sexual-abuse allegations against Polish clergy, entitled “Tell No One,” which has been viewed more than 22 million times. Victims’ advocates say the film...