Prepare both packs of vegan cheesecake pudding using non-dairy milk. Whisk until thickened and smooth, then let set for a few minutes.
Divide pudding evenly into three bowls.
Leave one portion plain (white), tint one portion red, and tint one portion blue.
Mix each color until fully smooth and evenly tinted.
Crush Oreos into fine crumbs using a food processor. Add a layer of Oreo crumbs into the bottom of each cup or jar.
Add a layer of one color of pudding and smooth gently.
Continue layering Oreos and alternating pudding colors until cups are filled.
Finish with a top layer of Oreo crumbs.
Top with red, white, and blue sprinkles just before serving. Chill until ready to serve.