Order of quantifiers in the definition of NP-completeness: does the reduction allow arbitrary polynomials?
The latter. The former is impossible for NP-complete languages by the nondeterministic time hierarchy theorem: assume for contradiction that $B$ is a language computable in nondeterministic time $p(n)$ such that every NP language reduces to it in time $q(n)$. Then
$$\mathrm{NP}\subseteq\mathrm{NTIME}(q(n)+p(q(n)))
\subsetneq\mathrm{NTIME}((q(n)+p(q(n)))n)\subseteq\mathrm{NP},$$
a contradiction.

You asked more generally about NP-hard languages. Consider the language $B\in\mathrm{NEXP}$ defined as
$$B=\{M\#w:\text{$M$ is a clocked poly-time NTM and $M$ accepts $w$}\}.$$
It is clear that any NP-language $A$ reduces to $B$ in time $n+c_A$ for some constant $c_A$, namely $c_A=1+|M_A|$ where $M_A$ is a fixed NTM accepting $A$.

But in fact, every $A\in\mathrm{NP}$ reduces to $B$ in time $3n+c$ for some constant $c$ independent of $A$. The reduction function reads up to the first $c_A$ letters of the input, storing them in its state. If the input has length $\le c_A$, the reduction directly finds the result in a lookup table, and outputs one of two fixed inputs that make $B$ accept or reject as appropriate. Otherwise, it outputs the description of $M_A$, copies the first $c_A$ input letters from the internal state to the output, and proceeds to copy the rest of the input. The running time is thus bounded by
$$\begin{cases}n+c,&amp;n\le c_A,\\c_A+2c_A+(n-c_A)\le3n,&amp;n&gt;c_A\end{cases}$$
for some absolute constant $c$ independent of $A$.