What would an exponential reduction from an NP-complete problem to P signify?
There would be no implications, and indeed I will now exhibit just such a reduction. The reduction takes an instance of vertex cover, finds the optimal value (in exponential time), and then reduces it to the language $\{0\} \in P$ (the language consisting of the single string $0$), by outputting $0$ if there is a vertex cover below the required threshold, $1$ otherwise.