this name was covered pretty recently on here...
But I'll repeat:
I don't think Primrose has a future as a name. It's just too theatrical, too cutesy and too overdone. Prim is not a particularly good nn as it's going to be associated with Prissy. Prim and Prissy, there's a fine line between being prim and being a priss.
I can sort of see it as a name for a cow or something, but on a person? No, I don't think we'll ever see much of that, thank goodness.
As for a full name for Rosie, well, I think Rosie sounds cheap and very dated and downscale. I'd expect to see a Rosie as a big fat red-haired woman with too much makeup on, swilling beer and shooting pool in a speakeasy.
I like the name Rose by itself. Lily- Rose is way too sickly-sweet and Mary Rose is not far behind it. Rosemary seems bland and dull to me. Rosa is pretty and more exotic than Rose.