I've done both. For Ashes To Ashes, I started posting the story before even starting the gameplay. I don't recommend that, by the way, since things can deviate from your plans very quickly.)
On the other hand, I didn't even start writing my next storylocke until the gameplay was done.
Thus, here's what I have to say about both approaches:
Writing while you're still doing the gameplay can allow the experience to be more raw, which is good if you're doing an emotionally-driven tale. Conversely, it's harder to plan character progression and things like that because you don't know who is going to be around for the next arc.
Writing after the gameplay allows for better planning, which enables better flow and, with good enough planning, less burnout. However, if you don't take good notes, there's the risk of losing the details that make a nuzlocke story unique (i.e: forgetting about a rival battle, tracking your feelings about that early game rodent from seeing it as a burden to recognizing it as a hero, etc.).
So my recommendation is to try to figure out which you have more of a problem with personally: planning or improvising. If you're better at planning, maybe wait to write (and especially post) your run until your gameplay is complete. If you're better at improvising, maybe having your gameplay progress align with your narrative progression will play to your strengths! And sometimes, a run demands one or the other.