I precisely read about your case. Here is my counseling.

Any custody case in Ukraine is complicated and the case where the one side is a foreigner who lives abroad surely is more complicated. However, there is a law (The Family Code of Ukraine) which regulates parents' rights and obligations.

First of all, it needs to say that now child already has a living place with her mother in Ukraine. Because during 5 years there was no any appeal from your side, it means it was negotiated about living place in Ukraine. To change her living place and send to stay with father, it needs to have a significant reason. Among reasons can be mother's offensive behavior against a child, immoral lifestyle, alcohol or drug addiction. But also it can be an impossibility to raise a child in the appropriate conditions (like don't care about child's health well, can't give appropriate education, no accommodation etc). Also, a court should not allow a child to stay with a parent who doesn't have own income.

In my opinion, if now your child still lives with her mother together, it would be too early to file custody case without significant reason. If the mother will leave Ukraine and start to work abroad and grandmother will care the child, as a father you will have a reason to file a custody case because you prefer to make your child live with parents instead grandparents.

According to The Family Code of Ukraine, a court can hand over a child to grandparents if none of the parents can care and raise a child well. So parents' care is preferable than grandparents care.

But be aware that the court will check all details about you. First, how you had been participated in raising the child before file custody case and if you give any financial support for the child. Where you live and if you have own income. Also, a court should check what is a child's behavior to you (now the child is very young but already can recognize people). Also, I would recommend you to attend a court at least one or more times to demonstrate your good will that judge can have an imagination about father because if the court will make a positive decision and send a child to live abroad they need to be sure that it's better for the child. Because the object is a child, a court doesn't have a chance to make a mistake.

Usually, the approximate duration of such case is 2 years.

About your right as a parent, I would like to explain that you have all right as a father. It means you have a right to communicate with your child, to have a meeting together, to participate in raising and making a decision about living, education, development etc. On the other side, to realize all these rights the best way is to negotiate with mother as you live abroad and the child stays with mother. You can negotiate it also through notarized agreement. If there is no another way, you can file a case about an order to communicate and meet your child.

Also, please be noted that a mother can require alimony from you as a father. According to the Ukraine law, all parents are obligated to care, raise and support their children.

So, probably in your case, the best way would be to make a good balance and suggest some support for your child. It will help to negotiate your participation in raising the son with her mother.

I tried to explain to you briefly all the ways where you can start. Please, be noticed that to deal through court is the most complicated way.

If you need any help or any further action, I would be glad to help you.