If all members of the old board resign simultaneously, what happens then? No more old board to agree to any new members. In a for-profit the shareholders can elect new board members, but in this case I don't know how it's supposed to work.

I've been privy to this happening at a nonprofit board. Depends on charter, but I've seen the old board tender their resignation and remain responsible only to vote for the appointment of their (usually interim to start) replacements. Normally in a nonprofit (not here), the membership of that nonprofit still has to ratify the new board in some kind of annual meeting; but in the meantime, the interim board can start making executive decisions about the org.