Mould/mildew spores can be removed by using pure lemon juice (from a fresh lemon). Squeeze the juice from the lemon and apply to the mouldy spots, then put in sunlight for a couple of hours and wash normally. Hopefully one or two treatments will do the trick. I have successfully used this treatment on antique linen. Bleach will remove the odour, but I have never successfully removed the stains using bleach (plus I worry about fabric dyes fading) but I am not familiar with the oxycleans/bleach in the USA, yours might work.
When you recover the table, make sure no mould spores remain on the surface that needs recovering...so out in the sun with it or gently rub in a mix a solution of oil of cloves and water - oil of cloves is a mould inhibitor. You should be able to get it at the pharmacy...a little goes a very long way. Let it dry thoroughly before recovering.
Let us know how you get on. Don't stress it happens to everyone somewhere along the line.
PS. Mould can have disasterous affects on human health, wear a mask when dealing with mould/mouldy areas. Mould spores are invisible. That in mind I would be inclined to wash the quilt before doing any more work....especially if you have allergies or asthma.