I renewed my second passport (still with validity, but full) by post just before Christmas. Didn't send in my other passport, but did mention it on the form and requested that it remain valid. Included letter from me explaining and also letter from company (as per requesting second passport initially).

Sent special delivery on Monday 19th, got text from passport office confirming receipt on Tuesday 20th, got cancelled old passport and new passport back Wednesday 21st ^

Sorry, can't help with the protocol re your second nationality passport, but renewing the second UK one - straightforward.