How to wire a 74HCT02 (NOR-gate)
As drawn, your inputs to the gate will always be low. The resistors R2 and R3 pull it low (to ground/the negative side of the supply). Then when you push the switches they connect the inputs to low again, so they have no effect. Rewire the switches so when you push them they connect the inputs to the high side of the supply, that way you will get low until you push the switch, then they will override the resistors and provide high.