Stephanie - Aw, I'm sorry to hear about this situation.
I can definitely help as I've removed lots of babies from hooded wicker nests.
Yes, you'll definitely want to remove the dead babies.
If they were small, you could have used tweezers to remove them. But now they're larger and it will be very difficult. It's tough even when they're small as it's often just a mass of body and it's hard to tell who's attached to what.
So here's what to do:
Remove the entire nest from the cage.
Get a spoon. I use plastic spoons from fast food restaurants since they're individually packaged and clean. The plastic is good because it's not cold. If you don't have plastic, just warm a metal spoon in your hands.
Wash everything - your hands, the spoon, etc.
Put out a paper towel on a table.
Place the spoon inside the nest and position it beside one of the babies. Tilt the nest so the baby rolls into the spoon.
Give him a second to settle into the spoon, then pull him out while sitting in the spoon and place him on the paper towel.
Repeat to remove the second one.
Now, remove the dead babies. You can use tweezers or a chopstick. They may be stuck to the nesting.
How old were they when they died? I'm guessing they were a bit older - beyond 3-4 days - since they're attracting flies. Usually if they're super small (4 days and under) they just dry up.
I think there's a high chance that bacteria-ridden body fluids have escaped from them. And that permeates the surrounding nesting. So remove the top 1-2 layers of nesting. (I use paper towels and toilet paper bits for nesting, so it's easy to see if it's clean or soiled. It's a bit more challenging with all natural materials like hay and grasses.)
I would have some additional fresh nesting handy in case you need to remove more than expected.
I always put the new nesting on the bottom and put a layer of the old (but clean) nesting on top.
Then, once the nest is clean, it's time to replace the babies. Put the spoon beside a baby and lift the edge of the paper towel to tilt him into the spoon. Let him settle and put him into the nest. Ideally, when returning him to the nest, position the baby so he's facing you and the spoon handle, so he goes into the nest bum first. Otherwise, they may extend their wings, making it difficult to get them back inside.
It may be a challenge to get them out of the nest, so just be prepared for it. Just be gentle and let them settle. That's key - giving them a second to settle in the spoon. If you try to remove the spoon with the baby too soon, you end up with legs and wings sticking out.
I've handled lots of babies and I've never had any problems, including in cases where I had to replace bits of nesting. There is a risk any time you mess with a nest, but usually, they're just fine.
Make sure any other birds or pets are put away when you do this. I've had my cockatiel go after babies.
It can help to have someone else to hold the nest and a flash light while you're working inside it. Or prop up the nest between two heavy objects (I use book ends.) I use a head lamp when doing it myself.
Best of luck! Let us know how it goes!