Clearly, since you are a British citizen, you are under no obligation to leave after 90 days and certainly could not be considered an illegal alien, let alone be banned or removed merely because you still find yourself in the Schengen area. Furthermore, 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 get stamps anymore nor have to prove when you entered or be asked any question about all this so that nobody will know or care that you did not have a passport at some point in the past.
The main question is what would happen if you encounter a police check 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.
Importantly, EU directive 2004/38/EC on the freedom of movement for EU citizens 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.
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 on you at all times is technically mandatory (the Netherlands), another one where holding an officially sanctioned form of ID is (Germany), some where getting a residence card is mandatory for EU citizens staying longer than 90 days and many where you are supposed to register your address with the authorities. You might therefore be liable for a fine, even as a British citizen, not because your stay is illegal but because holding some specific ID or document is mandatory in itself.
In practice, my guess is that your situation would be as much of a headache for the police as it is 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 with a verbal warning rather than go to the trouble of figuring out what to do with you. 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).
Finally, as you probably know, Switzerland is not part of the EU. It is bound by several association agreements with the EU, is part of the Schengen area and does not actively police its land border with France or run systematic ID check on air passengers coming from the Schengen area but it still has a special permit system for residents from other EU countries. I don't think the risk of getting in trouble is much bigger there than in the EU itself but I would still avoid it if at all possible.
