I have heard of birch water but I have no experience to share about harvesting it. In terms of the essential oil, this is often made with Betula lenta. Keep in mind, birch water won’t have the same properties as the essential oil. In order to make the oil, you will need to have access to a still and understand the specific process of distilling.
For the cherry juice, you should be able to use your own! The sourness would be a nice addition and you can always add some more sweetener if it needs to be balanced out.