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 $f>1$. 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 $\frac{f^2}f$ :)