Why was the examination delayed? Was it important for Evan's escape?

Actually not at all. Evans' examination got delayed since the Governor decided to check the invigilator McLeery so as to be sure that Evans could not use any article carried by McLeery to take him as a hostage to escape from the prison. Hence Jackson checked McLeery and took time on the issue of why he had a carried a semi-inflated rubber ring ot an exam. Soon, it came to be known that McLeery was suffering from piles and hence had to use it. Thus the exam got delayed as such.