Nanny is always just Nanny. She never has a name, and she never seems to have an identity outside of "nannying" the various Thropp children. But Nanny is doing a lot more in this story than just playing the role of governess/nursemaid.
Nanny pops up in every volume, except the third. Unbelievably, Nanny actually hangs in there till the last volume, when she is in her eighties. While other characters are dropping like flies, Nanny is the maternal figure who's always there, particularly for Elphaba. We see this from the very beginning of the novel, when Nanny is the only family member who is willing to accept and deal with Elphaba:
"Come here, you little demon. Nanny doesn't care. Nanny likes you." She was lying through her teeth, but unlike Frex she believed some lies were sanctioned by heaven. (1.4.39)
Mary Poppins she isn't. Nanny isn't without reservations about Elphaba; the child scares her at times. But Nanny never shows outright animosity towards her, and her mixture of reserve and affection carries on into their later relationship. This is fitting, since Nanny had a similar sort of prickly and bluntly honest relationship with Melena.
Since Melena died when Elphaba was so young, Nanny steps in and acts as an important conduit for these two characters. We really see this when Nanny starts telling Elphaba details about her mother after Elphaba is an adult:
"You needn't keep anything from me, dearie. Remember, Nanny was nursemaid to your mother too, and a more outgoing, sensual woman I have yet to meet. Convention didn't bind her, not in youth nor in married life." (126.96.36.199)
Nanny seems to keep up her gossipy tendencies throughout her very long life. The fact that Nanny has such a long life is quite significant. She represents old views (such as Lurlineism) that are being stomped out, yet she hangs on tenaciously for almost the entire novel. It's important that in the Wizard's changing and modernizing Oz, certain traditions (however false or even harmful) prove very persistent. The influence of the older generation, as represented by Nanny, lingers on. So it's significant that Nanny herself ends up raising all three Thropp children as well as Liir – she has a multi-generational influence.