IMPORTANT (17th JUNE 2021)
THE DEADLINE FOR APPLICATIONS IS 30th JUNE 2021
There is still some confusion concerning the need the to appy for the "new residency document"(card) for UK citizens covered by the WA particularly for people with Luxembourg or another nationality of an EU member state.
The Guichet (https://guichet.public.lu/en/citoyens/immigration/plus-3-mois/ressortissant-britannique/document-sejour-ressortissant-britannique-membre-famille.html) says that there are people who "are not required to apply for a new residence document" but that (if you read further) "they are free to do so if they so choose".
It is impossible to say what your personal circumstances will be in the future but one thing is certain, unless you are the holder of a 'carte diplomatique' , a 'carte de légitimation' or 'another residence document...' AND you continue to 'meet the conditions for obtaining such a document' then you do not need to apply for the "new residency document" now because 'they may apply to benefit from their rights under the Withdrawal Agreement' even after 30th June 2021. The problem is - this is trying to read the future and we don't have details of how this procedure will work.
HOWEVER:
If at some point in the future you find out that you need to prove that you are covered by the Withdrawal Agreement, having a little plastic card could make things a lot easier than going through a new procedure and having to request certified copies of documents etc.
The simple answer is - just do it!
The new residency card doesn't cost anything and
"what you have, you have".
CLICK HERE TO GO TO GUICHET.LU