Hi John,
Welcome to the wonderful world of electronics. It's like debating logic with strong-willed cats sometimes, frustrating but you get there in the end.
The resistor from pin 3 on timer 2 to the speaker looks like a 100 kiloOhm one to me, not 100 Ohms, big difference and it won't make any sound with a large resistor. And I'm suspicious about that huge polyester capacitor (red wine coloured one), I have 10uF, 100V that size; the 100nF, 250V are 1/10th that size. I doubt that is a 100nF capacitor on your circuit, unless it has a 10 zillion Volt rating
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Do you have a multimeter (DMM) or an oscilloscope? You need to start measuring voltages at the respective output pins to see what happens when a switch is clicked. Then repeat flicking a switch and measure what happens at pin 2 of the first timer. If not, but you do have some LEDs, you could put one LED on the output of each timer (as well as what's meant to be there) - in parallel, not series, to the actual loads - and see what lights up when you flick a switch. If they both light up, which they should, change the second timer's 100nF capacitor (from pins 2 and 6) for a 10uF to see if the LED flashes or not.
The two 100nF that are on the breadboard from V+ and ground to pin 8 and pin 1 of each timer are for decoupling. The circuit will work without them but it's good practice to include them.
I'm sure you are, but do take care to use the right value capacitor in its place.
Are you separating the other end of the diodes? They need to be in separate columns (the ones that are 5 holes long) on the breadboard.
How about trying with a different speaker? Do you have some old landline phone or something similar you can pull apart to scavenge one?
Maybe it's something to do with the battery, unlikely unless they're wornout, but still. Those batteries are all but useless when the voltage falls to around 7V, if you can measure them with a DMM. Try with a power supply, it can work from 5V up to 15V, so even the adapter from an old router or phone charger will be fine. Read the voltage and current rating on some "wall-wart" you may have lying around, they're usually 5V or 12V and from 0.5A to 1A. If you can't check the polarity of the wires, use a sacrificial LED and resistor to check which wire is + (5V/9V/12V) and which is - (0V) before frying your circuit with a reverse polarity accident.
What sort of 555s are you using? What is the prefix: LM/LMC/TLC/NE/SE/SA? Not all 555 cicuits will work with different models/versions, but most do. I've done this circuit with LMC555s (CMOS) and SE555s (TTL) and both versions work.