I am wondering if it is possible to cut the protruding part of the window sill here (see picture) such that my wall would be entirely flush. I am looking to fit a bespoke floating desk which would go right against the wall.
What are the risks here? Is the protruding part critical to the window stability?
Unless you get into either the mechanism or the seals of the window you can carve it up however you like. It's unlikely to be a structural concern.
Actually, that sill probably isn't part of the window to begin with. Jambs, stops, and sills are usually not part of the package. If it is part of the actual window, you may find a hollow which you'll have to cover with your desk or other trim.
If you decide to go forward, cut the caulk and/or paint all the way around that sill piece and give it a tap upward with a hammer. It may come free so you can remove it, run it through a table saw, and reinstall it. A fine bead of white painters caulk will have it back in place like new.
