Won’t happen. The Neandertal genome *may* be reconstructed in entire (though I very much doubt it), and eventually technology may appear which allows stored genome sequences to be reproduced into a double helix from a data set, but it’s not likely that such an approach would / could be used to produce a living Neandertal — other than those of us (a billion or two, or more) who are descended from Neandertal.
Then what do you think about all this talk about cloning a mammoth?