You have quite a bit of gain there and the op-amp you're using is far from being low noise. There are also opportunities for the ~200mA LED currents to cause issues or mains hum (though most mains hum should be rejected if the integration clock is correct). You are only seeing maybe 10-25uV referred to the input.
Start by determining the nature of the noise by using an oscilloscope on the outputs of the op-amps and looking to see if it's random looking noise or stepped when the LEDs switch.
Edit: Based on the comment, and from your video, from which I can see that your resolution is about 1mV and noise is around 20mV you should use a lower noise amplifier. You want an amplifier with noise in the 0.1-10Hz range that's maybe 1uV (or better). For example, the (quite old) OP07 is typically 0.35uV. You could also consider a more modern 'zero-drift' amplifier or an instrumentation amplifier. It's not enough to replace the gain-of-820 amplifier- the differential amplifier also is contributing noise. The final amplifier in your circuit probably isn't contributing enough noise to bother about.