I am a US citizen living in the UK on a Tier 1 visa. I've just submitted a passport renewal application because my US passport expires soon, and assuming there are no hiccups in that process, I should have my new passport in 10 working days. I understand that I need to report my change of circumstances and update my BRP within 3 months.

An opportunity to attend an event in Slovenia has arisen that will take place a few weeks after the arrival of my new passport, but I'm struggling to find information on how long the process of updating my BRP will take. If I am unable to update it before the trip to Slovenia, will I have trouble re-entering the UK with my new US passport and my current BRP?

Tangential question: may I report the change of circumstances and update my BRP at the same time or must I do one before the other?