ADC and Matched Filtering
Normally if you sample at Nyquist rate it's "good enough" but how does this interact with the discrete time correlator?


It interacts very well! See the following:


  Does sampling at a higher rate benefit the system?


You mean oversampling. 

If you filter your signal to the original signals bandwidth, which is now smaller than the Nyquist rate. You thus get a component that is correlated between samples (which is your signal) and one that isn't (which is your noise). By averaging / downsampling your signal, you increase the variance of your signal component more than that of your noise component – you get an SNR gain.


  Does up-sampling the signal and passing it through the correlator yield the same benefit? 


No, because upsampling doesn't "uncorrelate" the noise that you already digitize. 


  Is there any benefit to over-sampling and then discrete time matched filtering? 


Yes, as described above, the oversampling allows you to increase your SNR. Think of it this way:

Assume that if critically sampled, your pulse length is $N$; thus, your MF does a correlation over $N$ samples. 

You oversample your analog signal by a factor of $f&gt;1$. You interpolate your MF by the same factor, so that the pulse and the filter are now $Nf$ in length.

While the power of the samples stays the same, the amplitude of the signal part in the  (digital) correlator output has been scaled by a factor of $f$, which means the power  has been scaled by $f^2$. 

For the noise component in the correlator output: White Noise samples are uncorrelated, and we know that thus the variance of their sum (which is what a correlator is, weighted sum) is the sum of their variances. Since variance is power for zero-mean noise, that means we get a noise power increase of $f$.

Thus, oversampling by $f$ gets us an SNR increase of $\frac{f^2}f$ :)