Answer: C

Solution:

When the sector is rolled into a cone, its radius $10$ becomes the slant height, and its arc becomes the base circle.

The arc length is $$\dfrac{252}{360}\cdot 2\pi(10) = \dfrac{7}{10}\cdot 20\pi = 14\pi,$$ so the base circumference is $14\pi$ and the base radius is $7.$

The cone therefore has base radius $7$ and slant height $10,$ which is choice C.

Thus, the correct answer is C.