Your questions
Driving licence
I hold a valid driving licence issued in Luxembourg and live in the United Kingdom. Am I still able to use my Luxembourg licence after the end of the transition period (31 December 2020)?
Your Luxembourg driving licence will have to be transcribed into a British driving licence. Please contact the British authorities to find out the details of how this transcription can be done.
I hold a valid driving licence issued in Luxembourg and I plan to travel to the United Kingdom as part of my vacation or business trip. Am I still able to use my Luxembourg licence after the end of the transition period (31 December 2020)?
Yes.
Since the United Kingdom is, like Luxembourg, a contracting party to the Vienna Convention on Road Traffic of 8 November 1968, you will be able to continue using your valid Luxembourg driving licence while traveling in the United Kingdom, also after 31 December 2020.