This is an interesting question, and I would like to see other people's ideas on this. There may be discussion of this in the literature. The following are just my thoughts, working from first principles.
If highly non-Newtonian objects like black holes, naked singularities, and wormholes form through natural processes in our universe, then we expect them to form in runaway gravitational collapse. Once they form, they are extremely compact, and therefore we can't expect to resolve them in telescope images, except perhaps in the most favorable cases such as Sag A*. With wormholes, there is also the problem that they are expected to be unstable. Therefore, I would think that the chances of detecting a wormhole would be best if you try to detect its formation, not to detect it long after its formation.
For several decades, the standard expectation has been that the collapse of massive stars leads to the formation of a black hole. There have also been suggestions that it could lead to the formation of a naked singularity, and there have been serious proposals to observe such a thing (Joshi et al., "Distinguishing black holes from naked singularities through their accretion disk properties," apr 2013, https://arxiv.org/abs/1304.7331 ). One big difference between a black hole and a naked singularity is that a black hole is expected to radiate away only a small percentage of its mass-energy before the rest is trapped behind the event horizon, while in the case of a naked singularity it could be 100% radiated. There is also a high-frequency cut-off in the radiation spectrum when a black hole is formed, but not a naked singularity. I would think the formation of a wormhole would be similar observationally to the formation of a naked singularity in these respects.
There are theorems in general relativity that place tight constraints on the formation of wormholes by gravitational collapse. It requires a change in the topology of space, and theorems by Geroch and Borde say that you can't have topology change unless there are both exotic matter and violations of causality. (In technical terms, there has to be a violation of the weak energy condition, and the spacetime has to violate causal compactness.) Although the weak energy condition is violated by dark energy, we have no evidence that it is violated by any of the forms of matter -- typically baryonic matter and maybe dark matter -- that would be likely to contribute to astrophysical collapse. For this reason, if we are to have any chance of observing the formation of a wormhole by gravitational collapse, then we are necessarily talking about observing a process that falsifies classical GR in a regime where we had previously believed that it worked just fine. In this respect, it is very different from the observation of a naked singularity, which would be far out but not incompatible with GR.