Question

Which type of radiation does not involve the nucleus turning into the nucleus of a new element

Answer 1

Huh ? Lots of types but most obviously light or radio or whatever photon due to an electron moving from one orbit to another.

Answer 2

Damon has answered if the radiation is from outside the nucleus. I interpret your question as meaning radiation from the nucleus. Alpha particles and beta particles do change the original element into another one but gamma rays can be radiated and that does not change the identity of the element.