Oversampling while maintaining noise PSD
One important point you may be missing is assuming that the noise of the ADC is truly white over your sampling bandwidth- it isn't! You will have spurs from signal correlations and ADC non-linearities that will limit the noise floor you can achieve. This will be specified by the SFDR (spurious-free-dynamic-range) of the ADC for guidance as to what you may be able to achieve. Since the location of the spurs will be difficult to predict, you should assume that the strongest spur could be within your bandwidth of interest, in which case no linear filtering will reduce the power of the spur which exists at a single point in frequency.

That said, the filter will reduce white noise components, and the best metric is the EQNBW (equivalent noise bandwidth) for your filter; which is the equivalent bandwidth of a brickwall filter that would remove the same amount of noise. In this case the SNR improvement (processing gain) is 10Log10(2EQNBW/fs) assuming a low pass filter where EQNBW is the bandwidth from DC to the equivalent cutoff frequency.