As Miklo notes in the comments below, you can enable skipping the cutscenes using the cheat code of "N0CUT5" Note that the second character is a zero and the last character is a five. To enter a cheat code, pause the game, navigate to "Extras" and then "Enter Code"

If you would like to edit this option permanently into the game without needed to enter a cheat code, take the following steps:

1. Download a hex editor if you don't have one (I happened to use HxD: https://mh-nexus.de/en/ )
   
2. In the hex editor, open the GAME.DAT file from your \SteamLibrary\steamapps\common\LEGO Harry Potter\ folder. (Make a backup of the file just in case)
   
3. In the hex editor, search for:
   allowcutskip
4. Type over the text
   ;allowcutskip off
   with
   allowcutskip on
5. It is important that you just type over the existing text, starting with replacing the ";" with "a" and include the two spaces at the end -- you need to make sure you aren't adding or deleting characters, just overwriting.
   Here is the before:
   
   
   Here is the after:
   
6. Save the file.
   
7. After that just load the game and play it as you usually would, you'll just be able to skip the cutscenes.