Hello Mamam
I'm a newly-minted UK citizen / UK Yankee and am happy to answer your questions. I filed the application on 24 February and received notice of approval on 22 April. Processing time for citizenship applications aeems to have been relatively short recently - people who applied around the same time as I have been reporting similar processing time of two to three months. Hope it has stayed that way.
In my case, most effort went into finding suitable referees and, like Larrabee, getting details of past traffic offences etc. These need not be disclosed for visa applications but they do for citizenship. I went on a search through driving and court records that were available online for speeding tickets I incurred in my "reckless days of youth".
In the end I couldn't find much - the records have been expunged - and I had to resort to best guesses. There is a 500-letter comment section in the application form that I could use to explain what I did.
......
Perhaps the trickiest thing you should be aware of is not the application itself, but what come afterwards and how they would impact your future travel plans. Once your application is approved, you are required to attend a citizenship ceremony within three months. You become a UK citizen formally at the ceremony, and you'll be required to cut up your BRP and send it to Home Office.
Until the citizenship ceremony you can still travel as a permanent resident with a foreign passport and BRP. After the ceremony, you are supposed to carry a UK passport to travel in and out of the country (although people have been reporting success reentering UK with a foreign passport and the naturalisation certificate).
When you apply for the first UK passport though, you'll have to send in the naturalisation certificate and your current passport(s) to the Passport Office. This means there will be a period in which you don't have a valid travel document at hand. I'm in that limbo right now waiting for my first UK passport.
Finally (phew!): typically the UK Passport Office sends you the new passport before returning your supporting documents. From what I have been reading online, the time gap between the two could be substantial (several weeks). If you are a UK Yankee like myself, if you want to travel back to the US you MUST carry your American passport as an American citizen (the US is very strict on this). This could further delay travel plans to the US.
So - as I'm experiencing right now - your travel abroad will be "significantly impacted" for several months after the citizenship application. Be aware of that, and time your application / plan your travel accordingly.