It looks like just a couple of screws holding the sink to the wall. Maybe (hard to tell) a couple holding the pedestal to the floor. There's a lip where the sink sits on the pedestal, but I may be able to loosen the sink and pivot it enough to pull the pedestal out. Then secure the pipes (check for leaks!) and put the pedestal back. Will the two bolts hold the sink to the wall? Or will the whole think rip out.
I'd need to cut the caulk and probably paint that wall again. It doesn't look like there's a perfect tool to get into that little space. I was able to get the other drain out, so I'd like to think I can somehow get this tight enough. I think I used channel locks almost vertically and was able to generate enough torque to loosen the other one. Maybe I can somehow do the same to tighten the new one.
There must be some tool that would work though...