Clearly, since you are a British citizen, you are under no obligation to leave after 90 days and certainly could not be considered illegal, let alone be banned or removed merely because you still find yourself in the Schengen area. Once you have your British passport, the whole issue will become moot. You will be able to leave and enter the Schengen area with it, won't have to prove when you entered or be asked any question about all this.
The main question is what would happen if you are checked by the police inside the Schengen area in the meantime. Even if you are treated as an illegal alien all the way to a ban and removal (which I would consider very unlikely for a short overstay – even if you had no claim to British citizenship – as removing someone is expensive), you should logically be able to clear your name after the fact since you were a British citizen all along and as such entitled to stay even if you could not prove it at the time.
Still, you are in principle supposed to have a passport when staying in another EU country, if only to prove your British citizenship. There is at least one EU country where carrying ID at all times is mandatory and another one where holding an ID is. You could therefore be liable for a similar fine, even as a British citizen.
But EU directive 2004/38/EC on the freedom of movement contains language like
- Where a Union citizen, or a family member who is not a national of a Member State, does not have the necessary travel documents or, if required, the necessary visas, the Member State concerned shall, before turning them back, give such persons every reasonable opportunity to obtain the necessary documents or have them brought to them within a reasonable period of time or to corroborate or prove by other means that they are covered by the right of free movement and residence.
So as a British citizen, you enjoy extensive rights, even without your passport and you should be offered the possibility to prove your citizenship by other means if need be.
My guess is that it would be as much of a headache for the police as for you. In the unlikely event that someone asks for your ID, they would probably be more than happy to seize on any excuse to let you go. So I agree with @phoog that carrying your passport application, birth certificate or any other document that suggests you are British citizen would probably be enough to avoid any trouble (but don't act entitled, just suggest it gently while being ostensibly embarrassed about the fact you don't have the right document, it has always worked for me in similar situations).