Fortunately, the UV assignment of vertex tends to travel with it despite being compiled into a new ordering. Vertex ordering tends to be most important when it comes to morphs (also called blend shapes) since the description of how vertexes should move is compiled based on the assumption that both the original form and the new form have the same order and number of vertexes. From the UV Mapper screen grab you posted, it looked like the UV's came through intact, just overlapping.

There are three common approaches at this point, assuming you are happy with the unwrapping you have. The first approach to assign different materials to each group of non-overlapping UVs: For example skin_head, skin_body, skin_tail, skin_wings, etc. This way, you have different materials in Poser and can assign different color, bump, normal, maps, to each of these materials.

The second approach is to use a UV tool. I mostly use Modo for this, but there are other tools out there (notably UV Layout Pro from Headus). These have automated tools to "pack" the UV "islands" (each group of contiguous, flat UV mesh) so they don't overlap. It also will spin the islands to pack them optimally and scale individual islands (to give more or less resolution). You can them pack all your islands onto a single map so you only need a single material. I do not recommend this approach because the of the complexity of model and the loss in control of resolution and texture.

The third approach is to optimize your UVs in separate material groups as in the first approach, but then use a UV tool to stitch together some of the islands that may have been unnecessarily separated in the modeling process as well as fix up any overlapping issues. This is also a good time to do some UV optimization and flattening techniques. This is the most time-consuming approach, but gives the best results on complex organic models because you are in complete control of the placement, scale, arrangement of UV islands, as well as reducing the number of UV seams.

Cheers!

-K