I came to the UK as a US citizen, got indefinite leave to remain, and have just become a naturalised UK citizen. However, I am scheduled to travel abroad in a few days, and do not have time to apply for a UK passport. If I travel as scheduled, will I be able to re-enter the UK on my US passport (now without the biometric residence permit, which I had to destroy and send in when I became a UK citizen), provided that I travel again within 6 months? Or would I be turned away because I am not really a tourist?
Will I be able to re-enter the UK on my US passport (now without the biometric residence permit, which I had to destroy and send in when I became a UK citizen), provided that I travel again within 6 months?
Yes. You should bring whatever evidence you can of your newly acquired British citizenship. If you do not, you would risk being deported to the US. That is, consider what will happen if the immigration officer decides to look more closely at whether you are a "genuine visitor." During any extended interview, you will have to tell them that you are a British citizen. This will go more quickly if you have some documents to back it up.
The chance of that happening is rather lower these days because US citizens are now able to use the e-passport gates. If you use such a gate, you might be admitted without an interview.
Or would I be turned away because I am not really a tourist?
If you can prove that you are a British citizen, by any means, you cannot be turned away. If they find that you are not a genuine visitor, and you do not tell them about your British citizenship, and they do not find out about it on their own, then they will deport you to the US. This is extremely unlikely, of course, unless you actively pursue it by concealing your British citizenship.
Another possibility, of course, is that the e-passport gate flags you, sending you to an immigration officer, and the immigration officer does not investigate whether you are a genuine visitor to any significant depth. In that case, the officer will stamp you in with the standard six-month leave to enter, with employment and recourse to public funds prohibited. If that happens, the stamp means nothing. Because you are a British citizen, any purported leave to enter would be a legal nullity, as would any restrictions attached to it. If you get a stamp like that on your passport, you can just ignore it. You can remain in the UK as long as you want. You can also work or go on the dole.
Some countries, including the US, have a law that requires their citizens to use the passport of that country to enter. The UK is not such a country. A related question shows that UK immigration officers will even refrain from stamping your US passport if they know that you are a British citizen: UK border no longer stamps foreign passport if UK passport also held.