How long did the Omamaticaya live in Hometree?

[Human No More]

Written human history makes sense as a relative term. Either way from that remark, hometrees can be at the very least several thousand years old in the inhabitable size (if it is taken as written human history as opposed to the existence of human society), which means that they would likely take least a few hundred years before then in a growth stage.

I'm not really sure how they could be cultivated though - they are a single organism, as the different columns blend into each other, although they may well form symbiotic relationships with other plants, the tree itself certainly appears to be a single organism - this also creates the theory that one may have a lot of connections to other trees - while not as many as somewhere like a/the Tree of voices, they still act as important points in the network. If by cultivated, you mean that the Na'vi plant them in location in order to create one, then this might make a bit more sense, although with their likely growth speed, there wouldn't be much of a cause for this, as it would be hundreds of years at a minimum until it was of an inhabitable size.

A hometree is clearly not made of vines - it has branches and leaves, in addition to being made of wood ("you may make your bow from the wood of Hometree"), and as far as I know, vine species do not live that long, and most do not have the integrity to form a structure in that way. For a tree, there far less of a practical upper limit of size in the same way as with mammals, particularly in the environment of Pandora. A hometree does likely form symbiotic relationships with other smaller plant species, but I would say the core structure would till be a single tree.