Nature has invented the wheel three separate times that we know of, as far as I can tell.
1) ATP synthase, which is found (in slightly varying forms) in archaea, bacteria, protozoa, plants, fungi, animals - basically every living thing on Earth. It is older than eukaryotic cells, placing its evolution at more than 10^9 years in the past.
2) The bacterial flagellum. I couldn’t find data on when this might have evolved, but it seems likely that it was around 10^9 years ago. The split between archaea and bacteria was about 3.5*10^9 years ago. Archaea also have flagella, but they do not rotate.
3) Homo sapiens sapiens started building wheels probably sometime in the 6th millennium BC; it seems that people in Mesopotamia, the Caucasus, and central Europe learned this skill independently and at roughly the same time. The oldest known man-made wheel is the Ljubljana Marshes Wheel, found in Slovenia and estimated to be about 5150 years old; it is pictured below. Note that this wheel-invention required both ATP synthase and bacterial flagella (since humans are symbionts with both mitochondria and bacteria), so it’s not fair to say that nature has invented the wheel three times independently.