That there was increased levels of radioactivity above the background level found at Mohenjo-Daro.
This is unsourced and very vague. "In certain areas of that site, you find increased levels of radiation." In 4000 years, the Trinity site will only have slightly elevated levels of radiation. It won't make a Geiger counter go off-scale high.
Radioactive fallout comprises three classes of isotopes: short-lived products, with a half life less than a year, medium-lived products, with a half life between 4 and 100 years), and long-lived products, with a half life over 210000 years (fission). In 4000 years, the short and medium-lived products will all have decayed. That 4000 year period represents at least 40 half lives. The short and medium-lived products represent the vast majority (about 80%) of the yield of a fission reaction. The long-lived products emit radiation at a much slower rate.
That there was vitrification of the soil found at the same place.
No, but there were chunks of frit in the broken pottery found at the same place. One of the first things archeologists look for is where an ancient people disposed of their garbage, where they typically find lots of broken pottery. That pottery tells archeologist a lot about the level of technology of that civilization (CNN).
While ancient Indians didn't know how to make a nuclear bomb, they apparently did know how to make glazed pottery. Glazing is sintered glass created by placing a pot in a kiln after coating it with the right kind of material.
"Excavations at Harappa and Meohoaro have yielded a large variety of ceramic material which has been described as faince, vitreous paste, frit, composition, glaze, and mosaic glass." (Bhardwaj, p 28)
"Four pieces of glazed pottery found at Mohenjodaro are beyond all doubt the handiwork of a potter who was well-acquainted with the process of glazing and able to carry it to a high degree of perfection." (Bhardwaj, p 31)
fission: Fission product yield (wikipedia)
CNN: Trash or treasure? Sifting through ancient rubbish for archaeological gold (CNN)
Bhardwaj: H.C. Bhardwaj, Aspects of Ancient Indian Technology