Prescription antidepressants are not a single family of drugs, there are several with different mechanisms of action. Similarly gut health is a huge field. So it would not be possible to study such a broad topic in one or even a small number of scientific studies.
If you want to find out about studies on a specific prescription drug search PubMed or Google Scholar and/ or speak to your community pharmacist. Known side effects will be listed in the patient information leaflet (in the pharmacy box or online).
It is worth noting that various chronic mental health problems are linked to poorer physical health, in part because we do not always take care of ourselves. Targeted diet modification and physical activity are proven to be beneficial in mild to moderate depression, as well as in gut health. In some cases people can withdraw their prescription medication: speak to the prescribing doctor if you are interested in the self-care approach.