If I cast haste and give bardic inspiration to my Druid ally, will he be able to use these abilities if he then transforms using his wildshape?
The druid retains any abilities that their animal form can use.
Wild Shape states:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. (PHB, 67)
There's nothing that would prevent the druid from using Haste or Bardic Inspiration that has been given to them by someone else while in animal form, so they are able to maintain those effects and make use of them.