See the following for the entry requirements:
https://www.kcl.ac.uk/study/undergraduate/courses/law-llbFor your situation, I think the following should apply to you:
BTEC/CTEC Level 3 Extended Diploma with overall grade DDM and two A levels at A*A.
OR BTEC/CTEC Level 3 Diploma (120 credits) with overall grade DM and two A levels at A*A.
OR BTEC Subsidiary Diploma/Extended Certificate or CTEC Introductory Diploma/Extended Certificate with overall grade D and two A levels at A*A.
I don't know which BTEC you're doing, so you would need to check with your tutor, however, you would need at least A*A in your A Level subjects in order to get in.
If your BTEC is worth 3 A Levels, you would need DDM. If your BTEC is worth 2 A Levels, you need DM. If your BTEC is worth only 1 A Level, it's D.
If they didn't mention what the GCSE requirements are, then it's likely that they don't need your GCSEs - do check with uni admissions to confirm. However, I think the GCSE requirements would likely be reflected in your LNAT results anyway i.e. if you do well in your LNAT, you're likely the type of person who would have met any GCSE requirements they would have otherwise set. Do check with the uni to be sure though.