The fact that this question even needs to be asked is a reflection of the low standard of light aircraft training in the UK.
All UK flight instructors should be familiar with the contents of UK CAA Standards Document 19, version 9, November 2020, which covers the LAPL and PPL Skill Test (Aeroplanes) and provides policy and guidance for applicants and examiners.
From Standard Document 19, section 3.6.2:
"Although a private pilot would not normally give a take-off safety brief to a passenger, the applicant is expected to give such a brief to the examiner."
The document shows that the UK regulator, the CAA, assumes that all UK flight instructors are teaching the take-off safety briefing. One day the CAA may catch up, but don't hold your breath.
Even Birch and Bramson included this in their books 40 years ago, where they used the somewhat outdated phrase "Captains Brief," a term still beloved by legacy Flight Instructors.