Are you certain as to the identity of the aunt that the baby went to? And are we talking in England or Wales? Or somewhere else?
If it was in England and Wales, you might be able to find the aunt on the 1939 register - potentially the child would still be with the aunt and you might be able to work out any name changes that had happened. If the baby isn't with the aunt there, but you have located the aunt you could then use electoral rolls to track the aunt's movements. The baby might (when it was of voting age!) appear on the electoral roll with the aunt.
As soon as you have identified any name change, you should be able to track the baby forwards.
Or did you mean going the other way and finding the parentage of the baby?