I believe they all had wood propellers. Actually wooden blades set into carriers at the spinner. I'm not certain what the build process was, I recall a remark made in some documentation regarding the wavy surface finish on a D9's prop being due to the finish on the wood.
I suspect the reason was do to material technology and/or availability. The FW's props (all models) had a large cord width. I'd imagine an all metal prop would have been a significant weight in addition to other problems. A number that comes to mind on the Ta152s prop was 60-cm width.
Could be I am wrong, I'll look back into it...now you got me thinking about it. I know that some did, then so did some early mark spits and hurri's. Who knows, may have just been easier and cheaper to make that way. Germans are pretty resourceful with engineering tasks, do the most the least better than everyone else.