There was a post on the Embassy Facebook page covering this on 5th July 2021. It stated the following:
“When exchanging your licence, there should be no requirement to prove your residence in the UK when your UK licence was issued. If the issuing authority asks for this, please encourage them to contact the BMK, who will clarify the rules for exchanging UK licences.
If a regional authority is refusing to swap your valid UK licence or contact the BMK, please email the embassy, letting us know where in Austria you are and when you had your appointment email@example.com”