Okey, seems on my end it was a hardware issue (I tried hardware reset, disabling iPad features, removing apps - nothing helped. It looked like it helped but then touch delays were getting back). Apple replaced the iPad and all my problems went away.
I tried all solutions in this thread - nothing helped. After that I realized on my side the problem is connected to the Smart Cover. When I connect Smart Cover I have all sorts of problems with unresponsive touch. When I disconnect Smart Cover it works just fine!