Transistor switching problem
TL/DR: Your LEDs without resistors are diverting all the current away from the transistors so they are unable to switch.

In detail:

Let's take a quick look at just one small portion of your circuit in more detail:







simulate this circuit &ndash; Schematic created using CircuitLab

I have added an extra resistor into the circuit - Rw - which represents the resistance of the wires and pins of the LED, and I have given it a nominal 0.1Ω value. 

Now, let's say you set the GPIO HIGH, so it outputs 5V. The LED, D1, is a typical red LED with a forward voltage of, say, 2V.  Let's calculate the current through the LED.


I = (VCC - VF)/R = (5 - 2) / 0.1 = 3 / 0.1 = 30A


30A?! But the GPIO can't provide more than 40mA absolute maximum. So all that can come from the GPIO is being sent straight through the LED leaving nothing (much) for the transistor.

Oh and yes, all that current demand is doing unspeakable damage to the Arduino's GPIO pin.

The transistor's Base-Emitter junction has about a 0.7V drop across it, and, if it weren't for the LED, with the 1kΩ resistor that gives (5-0.7)/1000 = 4.3mA. Which with a hFE of around 100, would give a collector current of around 430mA. And that should be fine for most small relays.

But with all the current being dumped through the LED what can get through to the transistor won't be enough to turn on the relay. 

So increasing the resistance of the wire to the LED will decrease the current demands of the LED. And of course you do that by adding a resistor. So let's make that 1kΩ - a decent enough value for a simple indicator LED:







simulate this circuit

Now let's do the calculation for the LED's current:


I = (VCC - VF) / R = (5 - 2) / 1000 = 3 / 1000 = 3mA


Add that to the 4.3mA that wants to go through the transistor, and we have 7.3mA in total.

Now that is way below what the Arduino's GPIO can provide, so the IO pin is happy, the LED is happy and, most importantly, the transistor is happy.