Electronic – AVR ADC can measure rms

You over-engineered your module to the point that most people probably have real difficulty to understand, what you are trying to achieve. This how it can (and probably should be) simplified

1. This ADC is not really suitable for your application. You need something like AD7172-2 that can take handle (pseudo)-differential range of 5 V (with an external voltage reference). So search different manufacturers for suitable ADCa, ideally with integrated input buffers, but pay attention to input voltage limitations.
2. Delta Sigma ADCs have rather soft requirements for input filters. Passive filters are totally sufficient. That makes right opamp redundant.
3. Since you measure DC signal, your input filters are not as much about anti-aliasing but more about bandlimiting input noise , and do not need to be that tight. (see the link in 2. in case you still haven't)
4. Directly use differential inputs to measure your signal (assuming you followed recommendation 1), that's how they supposed to be used.Thus you get read of opamps that themselves can be the source of noise.

This is how your circuit should roughly look like

^{simulate this circuit – Schematic created using CircuitLab}

Note that the values of the resistors and capacitors are placeholders and do not represent recommendation for your case. They have to be chosen in accordance with master sampling frequency (of the delta-sigma ADC) and input impedance of the ADC. If there are no input buffers, filter resistors must be low valued (no more than 20 Ohm)

The above code gives a 20 ms delay. With a sine wave at 50 Hz, this should give the same value on every reading, since the period of such a sine wave is also 20 ms. However this is not the case and I can see no visible pattern in my results.

Hmm...

Am I missing something obvious here or could there be some other variables I am not taking into consideration? Any ideas would be appreciated.

A 20 ms intersample time is no good for 50 Hz measurements and yes, because the AC frequency is not constant (and neither is your clock) measurements made might be drifting in an out of phase seemingly randomly.

Look at this sinewave sampled at nearly the same period: -

It produces aliasing i.e. the blue output should be constant but slight differences in sampling frequency and sinewave cause the blue waveform to undulate up and down at the "difference" frequency.

If you want to properly measure RMS by sampling you should be sampling at around 1 kHz or greater especially if you are trying to measure the RMS of an AC current (harmonic distortions can be significant).