2 year old japanese lilac doesn't bloom
You've answered your own question. The two most common reasons that lilacs won't bloom are improper pruning and lack of light. In your case, based on these photos, these trees are not ever going to get enough sun to bloom prolifically.
You could try to prune the oaks to get more light in (but not now - never prune oaks between April and July). Or you could find another location for the lilacs that gets more sun. But after two years in the ground, I would not recommend digging them out and moving them.
Given the amount of shade you have, you might want to reconsider having sun-requiring lilacs as part of your landscape.