After G generations, each blob has a probability q=p^G of having a descendant. So, it seems to me that P will be distributed as a binomial with q and N as parameters.
The blobs don't reproduce with probability p in any given generation, they reproduce with probability p ever. The scenario doesn't require generations in the sense you seem to be thinking of, it could all happen within 1 second, or a first generation blob might reproduce after the highest generation blob that reproduces has already done so.
This thread is for the discussion of Less Wrong topics that have not appeared in recent posts. If a discussion gets unwieldy, celebrate by turning it into a top-level post.