I used model wastewater with rapeseed oil (popular oil) during biological treatment. The emulsion was prepared in an ultrasonic washer (2x15 minutes) and its lifetime lasted up to 24 h. Changes in the emulsions were observed on the basis of TOC analysis. You can prepare neem oil in the same way, but you must check temperature.
As mentioned by Ewa sonication might be a better choice to emulsify. Would go for high intensity sonication instead of using ultrasonic bath. Problem might be intensity of emulsification - vegetable oils in general are much more viscous than mineral oils. At least you have to use high speed rotor stator system.
Due to high content of unsaturated oils oxidation might be a further problem. Maybe antioxidants should be used.
Sediment you observe might be caused by solid contamination of the neem oil.
Try with high speed homogenizer (mixer) when you add the emulsion to water. I think that under any conditions (emulsifiers, oil/emulsifiers) you will have these separates layers.