The liquid nails is a great adheasive, but I've seen it take up to 10 days to set, at least when I do laminate counter tops that is :lol
I had the same thing happen recently too, the best way IMO is using support rods in combination with an adheasive, as someone else mentioned. If the the adheasie gives without the support of the rods, you get a sculpture "face plant". The rods will ensure that it at least stays in place.
Not sure about the goop though. I've seen goop melt down expanding insulation foam. Not sure what it will do to urethane foam. Whichever these busts are made of. I used a industry glue I got from a job, it's like a clear silicone caulk, but fast setting. Also used for countertops. Unfortunately there isn't a name on the bottle :lol
FP