While doing some work research I found this briefing from the UK House of Commons library which, I think, explains the situation around the requirement to travel on a British passport or with a certificate of entitlement more clearly than most.
These briefings are designed to support the activity of Parliament and I have always found them useful and up to date. Hopefully it I’ll help people working through this situation.
https://commonslibrary.parliament.uk/research-briefings/cbp-10344/
I'm just replying to "up" this useful post -- thank you for the link! -- except just for fun I'll mention that my husband late in life got an Irish passport in addition to his US passport. His mother emigrated from Ireland to the US, so the process was pretty easy for him. Of course, Ireland (Republic of Ireland) is a different country from the UK, so I wouldn't assume that any UK advice or regulations apply to his situation. -- EDITED -- please see a later comment from someone who knows much more than I do about how Ireland relates to UK when it comes to passport regulations.
Thank you for posting this, Emma. My husband is still going through the UK passport application process. I am sure there are a lot of people out there who don’t realize they are still UK citizens because they have been through the US naturalization process as my husband did.
Regarding Irish passport holders- the UK and Republic of Ireland have a common travel area and also freedom of movement (Irish citizens can live and work in the UK and vice versa). This also means Irish citizens don't need to have an ETA. So, if you are a dual Irish citizen visiting the UK, it makes sense to travel here on your Irish passport.
And just for additional info- even tho RoI is a different country to the UK, anyone born anywhere on the island of Ireland is eligible for an Irish passport- that includes people born in Northern Ireland.
