I wasn't sure either so I asked them at the feed store, and they said its a common misconception that most have about baby ducks. Ducklings CAN swim at 1 day old, wether or not they have their mum. It doesn't really have to do that much with the oils on their fuzz/feathers when they are this young. Well, at least not entirely. The oil gland doesn't fully develope on them until they are older.
Ducks have what are called 'air sacs' which keep them afloat. When they want to dive under, they pull their feathers in tight (which releases the air trapped beneath them), and expel all the air from their air sacs. They can then dive under the water. When they want to resurface, they kick towards the surface. As soon as their head is out of the water they take a deep breath which refills the air sacs and pulls them quickly to the surface. Then they give a shake, and all their feathers are returned to normal.
The fuzz on ducklings is totally different than a chicks fuzz, I am finding. It doesnt feel 'soft' or 'fluffy', but more 'fuzzy'. The water just beads up on them and rolls off. They should never swim when you can not watch them though until they are a bit older.
My duckies were already 4-5 days old when I got them, and they're now going on a week old. Yesterday I put them in a deeper tub to see what they could do, and they actually swam under the water! Then they bob back up and look totally dry. So weird!
And of course the chicks and older chickens think they are NUTS going in the water on their own! lol