Friedrich Nietzsche was probably conflating his own nihilism (and he WAS a nihilist, as even a cursory glance at his writing shows) onto Christianity and Buddhism.
Buddhism is rather self-effacing, but is decidedly not nihilistic. As one who has studied Buddhism and Christianity (though Buddhism far more casually), I can definitely conclude that neither is nihilistic.
Christianity is about filling your life with Jesus, especially after living according to laws (remember, it's a Re: to Judaism) fails.
Buddhism is about, if I understand correctly, also being a Re: but this time to Hinduism. Hinduism taught the universe as a sort of classroom where the soul develops. But Buddhism had marked differences (they were more closely linked to an idea of developing the self rather than belief in gods, probably seeing gods as distractions; they also believed in rebirth rather than reincarnation). The purpose is to kinda break free of the shackles that hold you to this Earth, while Hinduism is more about developing your soul. If I understand correctly, that is. Part of the problem is a bunch of sutras that I dunno how to really grasp.
Nihilism denies meaning or purpose to life, but both religions are very much purpose-driven. I would contend that my own religion is far more nihilistic than either of those, as I cannot answer the big questions satisfactorily as I am constantly seeking my own answers.
Buddhism does not deny that there are happy moments, bliss, joy and love in this life however what happens is you can not be happy, blissful and joyful constantly based on the transitory factors that condition these emotions to arise in this life so ultimately they turn out to become a form of suffering. Let me give an example.
Let's say something creates the emotion of joy for you and it lasts for days maybe even weeks. Ultimately what will dawn on you is you will begin to worry about losing this joy or if you are confident in it something may hinder it or get in the way of it's arising. You could grow angry at the source that is hindering your joy. Perhaps it is a person or an experience. You will become angry that this thing is disturbing your joy and ultimately your source of joy will turn into a form of angry or hatred.
Really the statement that life is suffering should be changed to a more realistic explanation such as, life is unfavorable. Does that still sound nihilistic? Well if it does then I have a bit of information for you to prove that Buddhism is NOT nihilistic.
Or put it more easily, constantly being happy usually involves boredom at some point. Boredom is a type of suffering.