When there's been no repentance in a person's life (they're an unbeliever) and then suddenly there is the beginning of repentance (they become a believer), it often appears to be quite a startling change. However, the fact that this happens for many does not mean that God works it at the same speed for all, or with every sort of sin in one's life; and the real change he does work may be the sort that's not as obviously noticeable. So there certainly may be occasions where it takes new believers some time to repent of some sins that would seem obvious to many of us. I've been a believer for decades and there are sins I still seem to have made little progress against.
I think this is particularly common in our current culture when it comes to sexual sin. The difference between the moral foundation in the church and that of the wider culture is so vast that it often takes new believers several months just to realize they've been sinning and to understand why their behavior is sinful. Then comes the matter of getting out of entangling relationships or changing deeply engrained habits, so that repentance in this area simply doesn't happen overnight. I've known several new believers who've had to struggle with this, and all of them took a matter of months, at least, to deal with God's call for repentance in that area. This doesn't mean we should excuse sin, but we should have some patience with a new believer who's adjusting to a whole new moral framework, and should deal with that person compassionately.