Travel before applying for British Passport

I have taken the Citizenship oath recently and have not applied for the British Passport yet. I still have my Indian passport, can I travel using the Indian passport at all. Is there a way to fast track my application for the British Passport, has anyone done this before. My wife is ill and we could do with some family help, maybe I can get the doctor to give us a letter. Would this help

There is no fast track option for first adult passports but the process takes a little less than 3 weeks in my experience, once you send the application off, you normally get an invitation to book an interview after a week or two, if you are willing to go to any available interview office, you could get a next day interview and provided everythings OK with the interview, you should have your passport 4 days later.

Thank you so much OfCharity, much appreciated.