What does Nathaniel mean?
From Hebrew Netan'el, combining natan 'he gave/given' + el 'God' — 'God has given'. A 'gift of God' name expressed as an act: God has given (this child).
What does Josiah mean?
From Hebrew Yoshiyahu, a theophoric name ending in Yah (Yahweh). Lexicons (BDB) read it 'Yah supports/supporteth'; the underlying verb (yasha) carries the sense 'save, deliver.' The popular rendering 'Yahweh heals' is common on name sites but is not supported by the root, so 'supports' is the better-attested gloss.