Thanks for your question about your Japanese tree lilac.These trees can tend to form a shrub, rather than a tree, and may need some pruning help to develop a leader and grow as a tree.
If there is enough flexibility in them, you may be able to encourage one of the existing limbs to become a leader. I can't tell for sure from the photo, but it looks like one of the limbs (not the strongest one) is generally trending upright. If so, then you need to prune the other limbs so they are shorter and less dominant than this one. The tip of a leader, or other branch, produces a plant hormone, auxin, which suppresses bud growth below it. If you make one branch the "top branch", it will grow faster than the others and become a leader. Later, when there is more growth, you can limb up the young tree to help if form the tree shape you want.
As to when it will bloom, I don't know for sure. All lilacs bloom on old wood (wood that grew the previous year), so once it starts blooming you should prune it right after it blooms. Until then, it will need to reach a certain state of maturity, but even if you leave it as s shrub shape, it will eventually bloom, when it is ready.
Looking at the photo, I am also concerned that there may be other problems. It looks very crowded by grass and other plants, which may be depriving of nutrients it needs to grow fast. It would be better to maintain a several feet wide mulched area around it.
Also, it MAY be planted too deeply - I can't tell from the photo. But if you followed the instructions on the tag when you planted it, (which are generally good instructions), the soil still may have settled and left it in a too-wet depression.