A private candidate is someone who
learns outside of a school / college setting, perhaps independently or via tutor, but who is
assessed within a school / college setting (i.e. they take their exams there). BTECs are primary assessed via coursework throughout the year, rather than via exams at the end of each year. As such it doesn't really match the private candidate model.
As I'm sure you're award BTEC is a brand owned by Pearson, with Pearson being the awarding body. As it says on their web site
here, "Subjects with 100% externally assessed components will be marked by Pearson and are available to private candidates." (By "externally assessed" they mean exams, which are marked by Pearson, as opposed to coursework, which is marked by exam centres.) They also provide details of accomodations for qualifications which have an element of non-examined assessments (e.g. the field-trip required for A level Geography, or the lab work required for A level sciences). The list of such qualifications / subjects is
here. It does not include BTECs.
I said above that, "I'm not sure about OCR Cambridge Technicals, but I suspect the same is true." I have just checked, and it is. As OCR say
here, "A private candidate is someone who takes an examination or assessment at an approved examination centre, but studies independently." They go onto say on
this page that "Internal assessment can be either coursework, controlled assessment or non-exam assessment" and that "Your chosen exam centre will be responsible for all aspects of the internal assessment, which includes supervision, authentication and marking." Whilst they don't specifically state that Technicals are not covered, the only qualifications they mention in this context are GCSEs, AS and A levels.
The Joint Council for Qualifications (JCQ), which is an umbrella organisation for the UK exam boards, has a web page
here where you can search for exam centres which accept private candidates by qualification and exam board. The main qualifications listed are GCSE, GCE (i.e. A level), and GCE AS Level. If you select the Other option, and then click-through onto the details page for some of the exam centres listed, you find that these "other" qualifications are things like "Edexcel International GCSE" or "International GCSE" - well, it was for all the ones I checked.