Circuit using only NAND gates
Your truth table, conversion to NAND gates and NOR gates looks correct, even the lower circuit looks correct, 

At this point I would turn to basic troubleshooting, bread boards are not perfect devices, so your likely facing connection issues, and the contacts in each hole gets worse and worse as its used more, combine this with logic chips that have probably been used 10 times before, so there is a risk that one of the inputs on a chip could be damaged

From your circuit you can for the deviations from the truth table work out what the output of each logic gate should be, go through until you find the one in disagreement, then correct the issue, be it a wire, a faulty chip, or a breadboard hole

Floating gates can be an issue for CMOS based gates, so using say 10K resistors to ground for the 4 external inputs can make sure you know all inputs are 0 until you set them high, its not likely to be the issue, rather can in rare cases be. 

Edit: I would not recommend your "OR" option on the schematic, it would short out the output of the logic gate through the LED, e.g. it tries to output 5V, the LED conducts somewhere around 1.5V, and the current increases, possibly pulling down the supply voltage of the logic chip.