Almost none of the applied math PhDs in the USA have BS degrees in applied math, because undergraduate degree programs in applied math are very rare (more commonly, one may have a "mathematics" degree but with an emphasis on applied math courses). A large proportion of applied math professors in the USA do not have a Ph.D. in applied math, because most of the applied math doctoral programs only came into existence in the last generation. For instance, my advisor's doctorate is in computer science, but his thesis was in numerical analysis. So the name of the degree program is not key.
I know people who have earned an applied math Ph.D. in the USA with undergraduate majors in engineering, physics, chemistry, computer science, and pure math. The non-math majors were from programs at very good schools with a heavy mathematical emphasis. As a professor at a university that is modelled after US universities, I have supervised successful students and postdocs whose backgrounds are in all those areas, as well as others in mechatronics and operations research. The transition for some was "quite harsh", but they persevered.
I think the quality and rigor of the program is an essential factor. A physics BS from a top school usually knows more mathematics (and can reason better in mathematical terms) than a math BS from a lower-tier school. Some computer science programs are, in fact, applied math programs; others involve very little math. And some engineering programs at lesser schools are virtually devoid of mathematics.
So the bottom line is that you need to know much more about a program than its name in order to determine if it will prepare you well.
I should add that I myself double majored in Physics/Astronomy and Math as an undergrad, then got Applied Math MS and PhD degrees.