Revelation 22:2

2 In the midst of the street of it, and on either side of the river, was there the tree of life, which bare twelve manner of fruits, and yielded her fruit every month: and the leaves of the tree were for the healing of the nations.

This is a very difficult passage. The correct translation is debatable here. I searched many sources and this is my best guess. The verse might say the tree of life  is in the midst of the river plain and extends to both sides of it. The tree is singular like in the Garden of Eden and it is not likely to be growing on both sides of the river. My opinion is that the tree is in the midst of the river and is sucking up the living water coming from the throne of God and it extends its branches and leaves over both riverbanks.

The translation also should not suggest that the tree bore a different fruit each month. The passage actually says that the tree bears twelve types of fruit and produces fruit each month. (In other words, 12 types of fruit are produced on the tree each and every month)

The leaves for the healing of the nations should be translated as  the leaves were for the health (or life) of the nations. A different translation of this passage is given below and it partly supports what I said.

Revelation 22:2 In the middle of its street. And on this side of the river and on that was the tree of life,  having twelve sorts of fruits, giving its fruit every month; and the leaves of the tree give life to the nations.  (BBE)

This is the same tree of life that was in the midst of the Garden of Eden. This physical manifestation of a spiritual truth will exist eternally to display that continual nourishment for the nations comes from God. The leaves for health represent that the spiritual life of all of God s people comes from the tree He has provided.

Since God mentions months in the eternal kingdom, it seems very likely that time will still exist in the eternal age.