In ancient Rome what did the Jewish community think of the persecution of early Christians?

Nero is over 200 years earlier than Constantine, more than a "few" generations. The short answer is that we have no idea. Jewish sources for the period are quite sparse and what there is makes no reference to Christianity until centuries later. The book of Acts in the New Testament claims that local synagogue leadership was usually hostile to Christian preachers and often tried to get them in trouble with local Roman officials. If that is an accurate portrayal, and was true at Rome itself (Acts doesn't say anything about Jews at Rome) presumably the Jewish community there was just as happy to see the persecution break out. But that would be pure supposition.