Juliet's Nurse is actually not present in the final scene in which both Juliet and Romeo die (V.iii.). That means that we will have to base our understanding of how the Nurse will react to Juliet's final death from her reaction to Juliet's earlier faked death. We will also be able to base an idea of how she feels about Romeo's death from her statements about Romeo.
Nurse took Juliet's initial faked death very hard and will probably take her second death even harder. Nurse cries and wails, frequently saying such things as, "O lamentable day!"(IV.v.33) and, "O woe! O woeful, woeful, woeful day!" (52). We may even be able to assume that she blames Juliet's parents for her death. Nurse reprimanded Juliet's father for rebuking her for proclaiming that she is not ready to marry Paris (III.v.174). Nurse probably blames Juliet's death on her parents for pushing her to marry so young. Hence, it can be said that Nurse will grieve significantly over Juliet's second death and blame Juliet's parents even more.
We may also be able to assume that Nurse will not care as deeply about Romeo's death; she may even place some blame for Juliet's death on Romeo. We know that Nurse never thought very highly of Romeo. After Romeo is exiled, Nurse encourages Juliet to forget about him and marry Paris instead (III.v.222-227). She even calls Romeo a "dishclout" in comparison to Paris and compares Paris to an "eagle" (III.v.229). In addition, after Romeo slew Paris, Nurse is very quick to call Romeo dishonest and to proclaim, "Shame come to Romeo!" (III.ii.94). Therefore, since we know that Nurse has never really liked nor trusted Romeo, we can assume that Nurse will not grieve over Romeo's death as much as she grieves over Juliet. It can also be assumed that Nurse places some of the blame for Juliet's death on Romeo.