German Umlauts distorted after RTF save with modified opentype font

I'm new to the forum and my question is a bit complicated to understand, even for me. I'll start at the beginning.

I was asked to make a modified version of a font for a client. The font was required to include a greek character set, which the original font did not have. So I created a range of greek characters and added them to the font file and exported as an opentype font. I'm a beginner with font creation, so I was a bit naive about this. I checked other fonts that DID have a greek character set for reference and tried to do it exactly the same (means I used the same glyph names and unicode designations)

Before I sent the font to the client, I thoroughly checked it on my mac (OS 10.6) and everything seemed to work.
The client however had problems, which prevented them from using the font at all. They have a bunch of RTF documents on their windows machine, which are mostly 500+ pages (so they cannot correct errors by hand.) They just select everything and changed the font from the one they were using before to my modified font.

The result was looking good, but after they save the file and reopen it, the german umlauts are gone and have been replaced by greek symbols (but even in a different font!)

Now I am not sure who is at fault here. I tend towards blaming their documents (word 2000) but it's hard to argue when I don't really know what is happening there. I tried to reproduce the error on my windows machine, but it didn't happen for me. Is there something obvious I could have done wrong that would produce this kind of error? I'd obviously be very happy if it wasn't something I did wrong.

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I'm sorry this got a bit lengthy. If you need any more information on my font files or anything, I'll post them right away. I did not know what made sense to include in this post here. Any help is greatly appreciated. And I'm sorry if I'm posting in the wrong thread. It seemed it had something to do with the font format. Thank you in advance.