Copper substituting silver in Tollen's reagent
For the particular reaction where you get a silver mirror with an aldehyde, I doubt it'll work for copper, due to the reduction potentials. I'll take acetaldehyde $\ce{&lt;-&gt;}$ acetate as an example here, but I believe that the reduction potentials of most aldehydes will be similar. The nitrate is a spectator ion here, so we need not consider its presence.

$$E^o_{\ce{Ag^+,Ag}}=+0.80$$
$$E^o_{\ce{Cu^{2+},Cu}}=+0.34$$

$$E^o_{\ce{CH3COO^{-},CH3CHO}}=+0.580$$

Our reaction involves the metal ($\ce{Ag}$ or $\ce{Cu}$ being reduced, so it must have a higher reduction potential than the acetate/acetaldehyde pair. Of the two metals, only silver has a higher SRP, thus the reaction involving silver nitrate is spontaneous--whereas the copper nitrate reaction is  infeasible.

Of course, you could find some substrate other than an aldehyde which reacts with copper nitrate to achieve the effect you want--though I'll have to look up some stuff to figure this out. I think the Tollen's reaction is a single step everyday redox reaction, which means that we just need to look for a stronger reducing agent which doesn't harm the glass.

Actually, it looks like the standard process for plating nonconductors with metals is to give them a conducting layer by dipping them in something, and then using electroplating. Not sure about this, though.