The UK one will have a page with a message along the lines of 'additional information: this person holds a [country] passport whose document number is XXXXX and which expires on XX/XX/XX'
If you have had to renew this passport, do you need to have this page updated in your UK passport?
The government website seems to say you only need to update your passport if you change your name, gender or appearance but I thought I'd double check just in case this may cause problems when traveling!
The closest I can find on the website is that you do not need to change your passport if you become a national of another country but it's not totally the same.