I've done this to my house- it's not fun no matter how you do it.
I would not remove your existing treads. You'll most likely find that your risers have dados cut in them for the tread- so if you remove them completely you'd never be able to exactly match the existing profile with your new hardwood.
I found that a multi-tool was great for getting into the corners when cutting off the nosing, but terrible for the whole job. I used a jigsaw and had great results. A quick sanding (the lumber was construction grade SPF) and you're ready to install the hardwood. You'll also need to fill in the dado from the tread with wood putty- I used the Rock Hard water putty stuff.
There are more things to consider than what's been listed already- such as the order of install, are you putting the risers on first (don't try to paint the construction grade wood that's there) or the treads (each has its advantages), are you ok with nail holes and putty showing or would you be ok with only using adhesive, etc.
Quick question- are you doing this to match the lower lever floor, or the upper level? Consideration needs to be placed on the difference between step heights. There's a very small window to work with, I'm thinking 3/8" difference max. If you've doing this to match hardwood on both levels, then you should be good (since you added the same height on both levels and you're raising the height on each stair tread everything will be the same height difference). You'll also need to make sure the hardwood you're selecting has floor nosing available as well.
I'll try to add some pictures of the process I went through. I don't remember if I have any construction photos, but I can at least snap a few pics of the final results- maybe that will help you with your decision.