I think you've got this a bit backwards; evolution is always the result of 'accidental' mutations. There's no reason that people in the north have to be paler than their southern relatives, it just so happens that that's an advantageous trait to have. It's very unlikely that it would simply not occur in human history, however it could have arisen later and been more isolated.
Mutations occur randomly every time an animal reproduces; each offspring is a combination of its parents genes with a handful of errors made. Any mutation is, in effect, equally likely, so the chance of first getting the light skinned gene is pretty darn certain over tens of thousands of years. Once you have one individual with a trait you move over to natural selection; if the mutated child has an advantage (which pale skinned people do in northern climates) then they will reproduce 'more' and therefore pass on their pale skinned gene to more offspring than otherwise. This continued until they supplanted the dark skinned population.
(I'd like to point out that based on current human behaviour it's entirely possible that the spread of the gene was helped by people finding the new skin colour attractive.)
In your hypothetical scenario everyone would indeed have dark skin, though differences in stature and facial structure would be more than sufficient for these people to distinguish different ethnicities.