I don't recommend doing what you're attempting to do but...
If you're covering 2x12 carpet treads, the 2x12 treads will need to be pulled off and have most of the overhang ripped off so you don't end up with a 2-1/2" or better overhang. Then reinstall the treads with liquid nails and screws or 16d nails.
From here, starting at the bottom, working towards the top, you'll need to secure stair nosing to the tread, possibly adding a return or two if you don't have boxed in treads. Now fill in between the riser and the back of the stair nosing you just installed. What you're thinking of doing is going to involve some work.
I've not done this with floating floor material, but I have done it with engineered planks as well as solid 3/4" prefinished hardwood.
Not sure of your location, but I can buy 4' oak stair treads for $25 each in Tennessee ( Lowe's ). 4' pine treads are a little cheaper, but the material is also softer. There's more work involved going this route and expense, but less safety concerns.
The reason my first sentence is, "I don't recommend this..." is because most stair nosing for laminate flooring is only 5/16" to 7/16" thick. If you drop something heavy on a stair nose or misstep, the nosing will most likely break.