Your questions
Driving licence
I live in Luxembourg with a valid driving licence issued by the United Kingdom. Am I still able to use my UK driving licence after the end of the transition period (31 December 2020)?
Yes, but only for a certain period of time. Until 31 December 2021 (final deadline), the valid driving license must be transcribed to a Luxembourg driving license. After this date, a test will become mandatory to obtain a Luxembourg driving license by transcription. Licenses that have not been transcribed by the end of this period will no longer be considered valid for driving a vehicle in Luxembourg.
Persons holding a British driving license who settle in Luxembourg on or after 1 January 2021 must have their license transcribed within 12 months (after having lived for in Luxembourg 6 months). After this period, the license will no longer be valid for driving in Luxembourg and a transcription requires the successful completion of a test.
I hold a valid UK driving licence and I plan to travel to Luxembourg as part of my vacation or business trip. Am I still able to use my UK driving 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 British driving licence while travelling in Luxembourg, also after 31 December 2020.