Tools that you will need: Circular saw, reciprocal saw, screw gun, and a drill.
First I would look around and access the electrical, plumbing, and other existing lines in the area or that might need to be re routed.
Secondly, I would get a tall ladder and begin to un hook the stairs from the floor joists of the bathroom. Try to carefully do this as not do damage the stair stringer or the floor joists. Then I would box in that area with joists and do the re flooring. Use those metal deck hangers for installing new joists in that are for the flooring to nail to.
Thirdly, I would find out how to remove the stair stringers from the basement flooring. Once done, you will need to repair that area with concrete or a hax saw to remove the bolts that supported the stairs.
Fourth, If you can, remove the stairs in one piece so that you can possibly swing the case around and mount it again on the more suitable area.
Fifth, now, you need to do some thinking. Because you are now ready to cut a new hole in the floor and also to remove at least two joists supporting the flooring. This needs to be done with some professional assistance so that the floor does not become weak and more problems happen. Here you need to duplicate what shoring or supporting was done for the floor in the old hole. You might need some heavy floor posts installed to support this area, since you are removing several floor joists. This is very hard to access without seeing the situation. WARNING: do be careful about what you do.
Sixth, Now, you can re install the old stairs, or better still make a set of new ones. You can use the template of the old as a pattern for the new. The stringers might be easier to copy this way than to re calculate.
Seventh, attach the stringers to the top with a series of carriage or lag bolts, attach to the basement flooring with bolts into the concrete flooring.
Install a good non skid surface to the steps and install a good durable railing.
Others will have some other good points to consider.