Craig Huneke told me about this [paper](https://www.tandfonline.com/doi/abs/10.1080/00927879708825958): "On the smoothness of blow-ups" ([MR1446135](https://mathscinet.ams.org/mathscinet-getitem?mr=1446135), [Zbl 0878.13004](https://zbmath.org/?q=an%3A0878.13004), by O'Carroll and Valla). The title alone seems to suggest it might be useful for you.