Both answers to this question seem equally reasonable to me.
If the answer is positive, I have no idea what the construction of such a space would look like....
If the answer is negative, I assume one would try to subdivide the cells somehow... but I don't really know how that would go.
I guess this came up because I was trying to think of an example of a CW-complex that wasn't homeomorphic to a Delta-complex... and figured the easiest way to make such a thing would be to make one that is not homotopic. This, of course, doesn't seem much easier to build, but at least easier to prove once you're done building.