The way that I fixed this is just connecting the Threshold pin (THR) to the ground. Leaving it floating caused problems even though I have seen it floating in other examples.

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Circuit 1. Full circuit.

• Initially D1 is on. V1 = 9V.
• Press SW1 the capacitor charges and the relay turns on, disconnecting the supply. V2 = 9 V. V1 = capacitor voltage which is falling. D1 turns off - probably by the time V1 reaches 8.5 V because it's anode is held at about 7 V by the current flowing through D2 and R1.
• The capacitor holds the relay on for a few seconds and the relay drops out.
• If SW1 is still pressed the cycle repeats.

Clarification: For current to flow in D1 the anode (top) will need to be about 1.5 - 2.0 V higher than the cathode (bottom). When the relay initially picks up V1 = V2 = 9 V so both LEDs light. Then C1 starts to discharge and V1 falls to 8.9, 8.8, 8.7 V. Meanwhile D2 is being powered directly from the 9 V battery. If, say, 15 mA flows through D2 and R1 the voltage at the top of R1 will be \$I \cdot R = 0.015 \cdot 470 = 7 V\$. But V1 is falling and at 8.5 V there will be (8.5 - 7) = 1.5 V across D1 and it will be very dim and shortly after go out completely.

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Circuit 2. Left LED only.

• Initially D1 is on.
• Press SW1 the capacitor charges and the relay turns on, disconnecting the supply.
• If SW1 is held the V1 = capacitor voltage. D1 gradually decays with C1 voltage until the relay drops out.
• If SW1 is still pressed the cycle repeats.

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Circuit 3. Right LED only.

• This time D2 turns on and gets full 9V.
• Relay hold-up lasts longer because C1 is only discharging through the relay coil. (D1 is missing.)

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Circuit 4. Independent LED resistors.

• This time V2 goes to 9 V as before and lights fully.
• V1 starts at 9 V and follows C1's voltage. In Circuit 1 D1 turned off quickly because it's anode was held at about 7 V by D2. This time it isn't so it remains forward biased with current flowing and light emitting!

Good first question. Some tips for future: There's a schematic button on the editor toolbar. It's quite easy to use. Draw your circuits with + at the top and - at the bottom. It's easier to see what's happening then.