I finally got Frances Jane Taylor up on GRO! Not sure if changing my browser from IE to Chrome had anything to do with it but I now get her up even just using the surname plus birth year and district. I see it cost 9 pounds+ for a full birth certificate, and that's including postage which would be around $15 Australian. When I order BDM certs. from NZBDM online they cost me $20 and that is for sending it via email and not a full one. I have ordered the birth certificate and hopefully will give me the name of the mother and if it states whether she was illegitimate. But even then it does not help me knowing whether Frances Jane was adopted by William and Frances Ward. I was hoping there was somewhere where I could find adoption records as trying to figure out why Francis Jane used the surname of Taylor, especially when she married John James Keefe and her father William Ward was a witness. I cannot find a birth registration for a Frances Jane Ward that matches the birth year as yet.
I have for Frances (nee Ware) Ward all her family information I could find while searching, mainly on ancestry when I was a member - birth, baptism, marriage and death registrations, census records plus info. on their trip out to NZ in 1856 for Frances, William and Frances Jane and Frances Ward's Will and Probate (found on NZ Archives) and NZ Electoral Rolls. For William Ward all I have when he lived in England is his marriage to Frances Ware and the name of his father, Joseph Ward who was also a sawyer.
Thanks again for all your advice and help. Much appreciated.