Most certainly not a Slider, but a Cooter - possibly a Pseudemys Nelsoni, as you have mentioned.
For those that don't know their turtles - to be a true Yellow Bellied Slider they have a yellow 'S' on the sides of their head and two black dots at the top of the underside of their shell. Red Eared Sliders (obviously
) have a thick red line on the sides of their head. Those that are, for simplicity, neither one nor the other, are classified as Cumberland Sliders.
Have a look at Austins Turtle Page, a US based reference site. Personally, I would say if the turtle could be put under a basking light where it cannot touch the bulb etc, with a temperature of 30c - 35c those scutes would come away, which is natural, leaving clean, fresh, healthy shell beneath. That said, given the conditions the turtle has been kept in with possibly no water filtration then there could indeed be infection occurring.
The turtle really needs much deeper water, uv, and a good diet plus ideally a safe outdoor pond. The top of the shell looks as though it has never even been in water.
Where are you based?