Thank you both for all your ideas so far. I don't think I can remove all the individual treads and risers, as they are wedged in with wooden wedges and glue from the back of the stairs, which I wouldn't be able to get at to properly secure the new treads/risers.
I could cut the nosing off (albeit with a lot of time and work to get around the curved risers), and then install new treads ontop of the existing treads, this would leave me with new, straight edged treads, but still a rounded riser, which I'm not sure would look good?
I was really looking forward to having a set of hardwood stairs when this was all done, but didn't know what I was getting into from the beginning. Currently my working plan, unless there's some other good suggestions, is to redo carpeting over just the stair treads in something more fitting to my decor, and then either a white thick veneer or wood thick veneer stained to match my upstairs flooring on the risers, and sand/paint the stringers white.