The 18-rated UK video release of this film was cut by 22 seconds by the BBFC to remove closeup shots of self-mutilation with a knife and a scalpel, and to reduce a scene where a man is repeatedly hit by a car.
Prostitute: What the beep are you doing?
Johnny: I'm gonna kill a bunch of people.

Put simply: Gore was removed. I don't know the full details but I do know that the scenes of self-mutilation were cut out when a character slices through his hand with a knife. Shortly after this his bloodily cutting a word into his arm was cut as well as his subsequent slicing all down his body with two knives. Quite nasty and similar cuts were originally made to Hellraiser 2 until a few years ago when it finally passed uncut. Like that film this would also now pass uncut. There may have been other brief cuts to gore in Bad Dreams by the way.