First, it probably depends on your DM. As Prestidigitation cannot duplicate the effects of a higher level spell, and the text of Diguise Self it reads, in part:
You can add or hide a minor feature or look like a completely different person.
Your DM may rule out that Wear costumes, at the smallest end of its range of effects, covers the effect of changing the color of the skin, therefore making use of Prestidigitation not allowed
In addition to that possibility, coloring the skin certainly seems to be within the range of what the spell could accomplish with its description. One face fits inside a 1 foot cube, and three separate foot cubes (for 3 rounds, or 18 seconds) could cover the face, hands, and feet.
To color all of your skin, you should focus on one section of the body that fits within a 1 foot per round cube. Generally speaking, a 6-foot-tall human body has two 3-foot-long arms, two 3-foot-long legs, a head that fits within a cubed foot, and a torso approximately 3-foot-tall and a just under two in width, but with hips and shoulders already included in the approximate measurements of the extremities:
o oooooo oo oo o o o o
That gives (very roughly) 15 sections of standing cube body. Therefore, coloring all the skin would take 15 rounds or 1 and a half minutes.
However, human skin actually has a very different color even on the same person, so such a costume would look strange only upon superficial inspection – they would have perfectly flat skin everywhere. On the plus side, this flatness could convince a DM that Prestidigitation does not duplicate or the slightest use of Wear costumes.