From the sounds of it, the 'mildew' you're seeing might just be the green paint underneath 'bleeding through'. I don't think you'd see mildew that quickly. And besides, if the paint was peeling now, then the causes may be from either the paint being poorly made, or from it not being mixed properly, or from it being applied when the trim was 'wet/damp'.
So, w/o seeing the job, itself, and going with my assumptions, I think you need to just do the prep work as if it wasn't ever done. If the paint is dried 'very well', then get a bristle-brush and some cleaner to 'prep' the trim. If there's any loose paint, the brush will remove it. Of course, after you've cleaned the trim, let it dry for a couple of days at least. Then, when it's good and dry outside, use a good latex Primer/Sealer. The Primer/Sealer will prevent the green from bleeding through to your topcoat. You may want to apply a 2nd coat of Primer/Sealer AFTER the 1st one has dried. (Green is famous for bleeding through.) Then, after the Primer/Sealer is dry, go ahead and paint on 2 layers of topcoat.
Now, to reduce your chances of 1) having to do this job again in a few years, and 2) getting the desired results, visit your local Paint Retailer (which is/are found in the YELLOW PAGES under PAINT - RETAIL.) There, they will guide you with proper paint and tools. (It's what these folks do for a living.) In short, you'll spend good $$$ on good stuff. I say this especially since you're already having to do this job a 2nd time in a very short order. As usual, BEFORE you leave the store, read the Instructions on the can so you can ask your questions with the 'experts' inside the store. So be sure to go there when you're NOT in a hurry! Other than that, if you have any other specific ?'s, post up. If you can wait a couple more days, perhaps others will have some more info to offer.
My best to ya and hope this helps.
Jay J -Moderator
PS: I don't think this is the type of job that has to be FORCIBLY done in the Fall (if it's getting cold where you live.) Primer and Paint have specific humidity and temperature 'ranges' in which they're to be applied. They have these 'conditions' to ensure their product performs as advertised. (Too much humidity, for example, can cause blistering, and direct sun will not properly dry the paint (to name a couple of reasons) ...)