I try to avoid washing "flimsies" if I can and work with them first. There have been things that needed sanitizing before working on, just treat them 10 times more carefully. If you don't feel a noticeable difference in the yellowed areas, I'd finish working on it first.
You can soak the finished product with something oxyclean or orvus or many other things. There are many different favorite treatments.
I recommend putting the (finished) quilt in a plastic laundry basket (the cheap basket type, you want big holes), and then put the basket with the quilt in the bathtub. I usually start with water as hot as mine gets (which is above recommended settings for households with children) and dissolve whatever I'm using first before putting in the textiles. The basket helps keep the strain/weight off the quilt as you gently swish it around, and it allows you to actually get in and stomp on it to get the excess water out.
Even with modern workmanship and back, still treat the entire top as if it is as delicate/old as the top. Don't ever pick up a delicate wet quilt by just a corner, the weight and pull can be tremendous.