We are not told what nationality Bathsheba was or had. However, we have the following:
- Bathsheba is a Hebrew name meaning either "daughter of an oath" or daughter of the seventh (an allusion to the Sabbath)
- Bathsheba Eliam (2 Sam 11:3) or Amiel (2 Chron 3:5), which, because of the way Hebrew works is the same name. These are also Hebrew names.
- Bathsheba's father, Eliam, was the son of Ahithophel (2 Sam 23:8), also a Hebrew name.
The above data strongly suggests that Bathsheba was both Hebrew and descended from Hebrews, but there are other people with Hebrew names that were not Hebrew such as Uriah himself (a Hebrew name meaning "YHWH is my light") who was Hittite. However, such exceptions are usually noted in the text (as per Uriah) which is absent for Bathsheba.
On the balance of evidence, it is highly probable that Bathsheba was a Hebrew girl.
The above conclusion should be tentative because of the difficulty in even defining what a Hebrew is because Boaz' mother was a Caanaite woman but was still counted as Hebrew; the same is true for Moses' children whose mother was also not a biological Hebrew; the same is true of Judah's children whose mother was Tamar, etc, etc.