From The Ministry of Foreign Affairs, we learn that:
The attestation of being registered in the registry by 31 December 2020 (see below) or the self-certification of residence is the ordinary means for obtaining the issue of the electronic residence document by the Police Headquarters of the province of residence.
However, British citizens who are not yet registered in the registry by 31 December 2020, but who demonstrate, through appropriate documentation proving a legal stay on the national territory on the aforementioned date (e.g. employment contract, certificate of enrollment in a course of study) may request the issue of the electronic residence document at the police headquarters of the province of residence.
You can request an appointment to Questura By e-mail (see above)