Take off the case and unplug the circuit board that goes to the front buttons. Check and make sure he little nub on the start button didnt break off. Then re-insert fully and try it again. With it all off try to hit the button on the circuit board without using the actual button to see if it works.
IF you replaced the screen you may have accidentally unplugged the wires from the UMD drive. I have completely disassembled 5 PSPs and the UMD cord comes through a hole from the back to the front and connects right under the screen. Check that.