Die Tierwelt = fauna. A limited (in time or geographical) variety of animals. This one is probably closest to meaning wildlife, but mostly also the plants should be mentioned.. (Die Tier- und Pflanzenwelt / Die Natur)
Das Tierreich = animal kingdom. (Animalia, taxonomy)
(Yes, I've reported it)
Depending on the definition, "wildlife" also includes plants. For example, Oxford Dictionary says "fauna (and sometimes flora)". "Tierwelt" sounds more explicit in including only animals. I'm not an expert though, so I'd ask several German speakers and see what the consensus seems to be.