Oversampling will not do anything to change the existing noise in the waveform other than the quantization noise that is added. Sampling does not change the waveform but the quantization process of sampling with ADC’s will add additional noise to it (quantization noise and other additive noise collectively given as the “Equivalent Number of Bits” or ENOB). The OP is referring to the Gaussian noise that may be sampled, which itself will not be modified other than the additional noise added. This makes complete sense since our goal with sampling a waveform is to represent it with minimum distortion; there is nothing different between noise and a waveform of interest that would somehow be sampled differently by the ADC.
With regards to the quantization noise, oversampling will spread the additive quantization noise out over a wider frequency and thus reducing it density on a per Hz basis, since the quantization noise is well modelled as a white noise over the sampling bandwidth. The total quantization noise power doesn't change, but the noise density would be lower since that noise is spread over a wider bandwidth. When the ENOB is limited by spurs, then further oversampling will not reduce the noise further since a narrow band spur is confined in frequency rather than a spread noise. Further as sampling is increased the ENOB typically reduces due to non-linearities and aperture uncertainty in the ADC front end.
Given this, the typical goal in a system is to have sufficient gain in front of the ADC such that the amplified thermal noise floor is higher than the added noise from the ADC with sufficient margin such that the noise floor we observe digitally is the thermal noise floor of the analog waveform and not the added quantization noise of the ADC (for the latter case we would then be swamping out the signal with additional added noise which is typically not desirable when our interest is SNR). Oversampling can help meet this goal by effectively increasing the number of bits in the ADC (increasing its dynamic range).
For further details please refer to: What are advantages of having higher sampling rate of a signal?