What causes this acorn disease?
The structure on the oak nut is a knopper gall.


  It is a distortion of growing acorns on pedunculate oak (Quercus robur L.) trees, caused by gall wasps(Andricus quercuscalicis), which lay eggs in buds with their ovipositor.




An Andricus quercuscalicis larva in an open gall of an acorn:
Link 

Here's a video by BBC on the same you may find it interesting.