Hi,
Have you tried looking for the birth entries of Rose, George or your grandfather Samuel?. The free BMD is not going to list them
http://freecen.rootsweb.com/index.html( I believe it only goes up to 1910). And then you can order one of their birth certificates to get the mother's "Rose" real name and address and where she was living at the time.
You can either search the record offices in the UK ( someone else will have to advise you on that, I'm not in the UK) , or use the pay per view 1837 site
http://www.1837online.com/Trace2web/aboutuslisting.jsp( which I have had to use for a lot of my family ) to get the volume and page number, to order the birth certificate
http://www.familyrecords.gov.uk/.
I'd start off first by trying to find your grandfather samuel's birth entry. If you can find it order his birth certificate to get more information about his mother ( and father) . Samuel's ( or his siblings) birth certificate will tell you his mother's ("Rose") maiden name and her christian name, and where she was living at the time of his birth.
Hope that is of some help to you

P.S don't assume Samuel's mother may be called Rose, it could be his grandmother's name , his mother or father may have called one of their daughters after their own parents. Or an an aunt, anything .Or even because they thought it was a pretty name, no reason at all sometimes.
Try looking for your grandfather Samuel Greenwood birth entry first if you can find it, and order his birth cert to find your "Rose"