The following represents my personal thoughts and views, and not that of any particular Christian tradition (although it would find a home in several). Indeed Adam and Eve were by nature physically mortal, just as all living things were (and are) mortal. The tree of life, like so many objects in Scripture, is represented as a sign and symbol of the everlasting life which Adam received immediately from God—as will the glorified saints in the new heavens and new earth (Rev. 2:7; 22:14); as observed in Vincent's Word Studies, "To eat of the tree of life expresses participation in the life eternal."
But ever since the fall, Adam and his posterity were cut off from the tree of life and no longer received everlasting life immediately, but rather mediately through Christ (1 John 5:11). Thus Jesus Christ testifies that those who hear his gospel and believe have eternal life and will not be condemned, but have "crossed over from death to life" (John 5:24). Give attention to that phrase, "from death to life." Immortality is something which in Christ we seek; and if we seek it, then we do not now possess it (Rom. 2:7; 1 Cor. 15:53). Likewise Paul speaks about how "sin reigned in death" among mankind (Rom. 5:21), and how could sin reign but among those who are physically alive?
Rather, death is understood covenantally in a spiritual context, which is how sin reigns in death for those who are physically alive but spiritually dead in Adam, whereas grace reigns through righteousness to eternal life for those justified in Christ, such that we spiritually live even though we physically die. (Cf. John 11:25-26, "The one who believes in me will live even if he dies, and the one who lives and believes in me will never die.") As John Calvin noted, "Let us know, therefore, that when we have departed from Christ nothing remains for us but death" (cf. Rev. 20:6, 14).