In this problem, "runtimes" refer to worst-case complexity compared up to constant factor.
Say you have two problems, A and B, in the same time hierarchy, and it is clear that algorithm P solves A with that time complexity.
Suppose an oracle spits out the result of problem A using algorithm P. Then does problem B, with the help of this oracle (call this B^P), necessarily have an algorithm running in less time than problem B without the oracle?