If you want to do physics research you realistically need to do a physics degree. There are some areas of applied/experimental physics where an engineering background in either materials or electronic engineering might be suitable, but pretty much any other engineering discipline (except maybe chemical or the extremely rare nuclear engineering) isn't going to be suitable preparation for that, and also the areas those degrees prepare you for are not related to the areas you have expressed an interest in.
That said, I would note "quantum or nuclear" physics covers an enormous range of research areas, both theoretical and experimental, basic and applied. So it's not really that meaningful I suppose? Even something like "I want to go into quantum theory" still leaves quite a lot of scope. "Nuclear" physics ranges from applied nuclear physics as relevant to nuclear engineering etc, to the experimental but definitely not "applied" fusion research, to fundamental theoretical research in high energy physics.
However, given you are interested even very broadly in those areas, the only degree which really suits is, as stated, physics. So I'd suggest you pursue that as a degree. Going from a physics undergrad to a career in the engineering industry is much more straightforward and typical than going from an engineering undergrad to physics academia.