My interior staircase has a closed stringer and I want to change the first step to add a bullnose tread. Can I cut the stringer: is there any way this can be done?
Peel back the side of the carpet just enough so you can see if the tread has been dadoed (sits in a trough) into the stringer. If not, the tread is likely just nailed onto the riser and can be gradually popped off using a hammer and a block of wood under under the tread lip. If the next riser is on top of the tread rear, the nails holding the two parts may have to be cut with reciprocating saw or sabre saw (if you're good!) Can't say where the nails will be, locations vary.
If the tread is in a dado on both sides, a piece will have to be cut from the middle front to back and the three pieces pulled out. Your stairs look like typical stairs intended for carpeting. Often, what looks like a stringer is a trim plate slid between stairs and the true stringer and disassembly is easy. The outside has an identical trim plate running top to bottom.
It can be done, but it may not turn out like you think. Bullnose treads are usually part of a open riser setup. The newel post are typically in the middle of the tread, in the center of the radius of the bullnose. I guess is there no rule it HAS to be in the center, so to get that detail the upper rail, although it is not in the picture so I can make a good call on it, needs to be shortened and the newel relocated. No big deal while the alteration is going on, but you should be aware it should be more than simply adding a bullnose tread