Can we derandomize subexponential algorithms given P=BPP?

Under $BPP=P$ conjecture randomization does not have much power for poly time algorithms.

Can we say the same about randomized subexp algorithms like number field sieve?

Derandomization results transfer "upwards" using the technique of padding. Assume $\mathsf{P}=\mathsf{BPP}$. Suppose that some problem $A$ can be solved in randomized superpolynomial time $t(n)$ (which is time-constructible). Let $A'$ be the same problem, but with inputs padded to length $t(n)$. Then $A' \in \mathsf{BPP} = \mathsf{P}$. The following algorithm then solves $A$ in deterministic time $t(n)$: pad the input to length $t(n)$, then apply the polynomial time algorithm for $A'$.

Conversely, it is possible (given the current state of knowledge) that subexponential time algorithms can be randomized, but polynomial time algorithms can't (and it might be possible to construct a relativized world in which this happens unconditionally). That is, results don't transfer "downwards".

StackExchange Computer Science Q#55358, answer score: 3