I have a UserControl that is used in a parent control in this way:
Once you set the DataContext your bindings will use it, so I would expect this behaviour -- looking for SelectedEntity.Name on the TranslationTextInputViewModel.
There are a few ways to get this working. Personally, I like to model these relationships in the view-models themselves (view-models with view-model properties), but in this situation I'd probably try this, as unpleasant as it feels: