In Chapter 1, Batman Takes Over, Batman and Robin are in the Bat Cave still dressed in their caped outfits while examining evidence. Alfred phones to say that Vickie has arrived at the residence to see Bruce Wayne. Batman changes into his business suit while Robin does not change clothing and remains in the cave to examine evidence. Bruce then tricks Vickie by having Robin (still dressed in his cape in the cave) phone upstairs with the message that Vickie must leave to cover an auto accident. When Bruce returns to the cave, Robin is now dressed back in his business suit like Bruce, so they both pull out their caped outfits from a drawer to change together in order to leave and fight crime.
For three chapters, Robin tries to persuade Batman that the professor can't be the Wizard because he's a wheelchair invalid. In the 14th chapter, we see the professor up and walking about, even in front of Batman and Robin, with no explanation whatsoever, and no questioning on Batman and Robin's part.
When the gang lands their plane in episode # 1, Batman Takes Over, there are windows in the nose of the plane. When the gang takes off in episode # 2, Tunnel Of Terror, it is a different plane, with no windows in the nose.