Hello! I applied for and received citizenship last year (2022). The citizenship application is not like the visa applications. You don't send off your documents, you just upload them so you retain your passport throughout. You can also travel while the application is processing and up until you actually receive your citizenship, as your status in the country is not affected while it is processing (you retain ILR). It is only once you actually receive citizenship by attending your ceremony and receiving your certificate of naturalisation that travel becomes complicated, as you will need a British passport. Now, people have been successful in flying back on their American passports with their naturalisation certificate (usually get pulled aside while they check things, but will let you in).
Anyways, long answer short- yes, you can travel during the citizenship application. Good luck!