For citizenship applications – yes. Once your British citizenship application has been submitted, you can travel outside of the UK while it is being considered. As a part of your application, you can submit a copy of your passport and keep the original in order to travel.
Your application will not affect your immigration permissions, but you will need to demonstrate entitlement to re-enter. You should travel with evidence of your relevant immigration status.
Keep in mind that you need to enrol your biometric information within 45 days of your Naturalisation application, which may be necessary in person. Failure to provide this will nullify your pending application.
You will also be required to arrange your British citizenship ceremony within 90 days of your application being approved.
In short, it is permissible to travel whilst your citizenship application is pending, but in practice you need to be ready to respond to the next stages of the citizenship process, which will normally require some physical attendance in the UK.
Note that the position is very different for non-citizenship applications made from within the UK, i.e. visa applications for permission to stay. Paragraph 34K of the Immigration Rules, states that if you leave the Common Travel Area whilst an application for permission to stay is still pending, then the application will automatically be treated as withdrawn, on the date you left the Common Travel Area. The Common Travel Area includes the UK, Ireland, Jersey, Guernsey, and the Isle of Man. It goes without saying that allowing your leave to remain application to accidentally be treated as withdrawn, whilst outside the UK could have devastating consequences for your immigration status. As well as losing your right to re-enter the UK and your continuity of residence, you may need to re-apply for entry clearance from outside the UK.
