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Would the mother consent?
You will have to apply to the court for a court order. And that would require serving her personally. But, if you cannot then you will have to get an order for what is called substituted service. What this means is that you have to prove to the Court that you tried your best to find her and if you can prove that then the Court will allow you to serve her through an alternative way, such as serving her family, advertising in the local paper, social media etc.
But, you do need to be able to show your efforts to locate her or the Court will not consider your request.
It is very, very likely you will succeed in getting an order for sole custody. The mother has abandoned the children completely and couldn't possibly be considered someone who should have any rights with respect to decision making.
Your case is very strong and your children are very lucky to be able to have a strong family life despite what their birth mother has done.