Re: which primer?
First, it sounds to me like you've got a severe moisture problem that needs to be addressed before you attempt to repaint with anything or you're just inviting all the original problems back. Invest in an exhaust fan, if you've got one, then A - is it large enough, and B - is it properly ducted to the exterior of the home and not just venting to the attic.
Second, whenever you wash walls down with anything, you always follow up with a couple of good rinses with clean, clear water to remove any chemical residues that can hinder paint adhesion - that includes bleach residues. The saving grace here is that it sounds like you've floated out the walls with drywall compound, which will isolate any residues from the paint, so it should be fine.
Third, no special primers should be required. I've remodeled many overly moist bathrooms, rectified the moisture problem, repair the walls as you've described, and used standard primer, typically Zinnser Bullseye 123 (blue label ), and semi-gloss paint. An alkyd paint is a little harder than latex so that if / when mildew returns, it is more stain resistant, but alkyds will stain and propagate mildew just as effectively as anything else if your moisture problems are not addressed.
I suffer from CDO ... Its like OCD, but in alphabetical order, LIKE IT SHOULD BE!!!