I have (at last) found some specific information about EU/EEA cross-border prescriptions post-brexit - there is a special case in respect of Ireland.
If you have a medical prescription from a doctor in the European Economic Area (EEA), the prescription is valid in all other EEA countries if it contains certain information. The EEA includes the member states of the European Union as well as Iceland, Liechtenstein and Norway.
A medical prescription written by a doctor registered in the United Kingdom (UK) continues to be valid in Ireland, if it contains certain information. Likewise, a medical prescription written by a doctor registered in Ireland continues to be valid in the UK, if it contains certain information.
See 'Rules' below for the information that must be included.
A medical prescription is not valid in Ireland if it is issued by a UK-registered doctor, via an online service, to a person living in Ireland.
REST OF THE WORLD (GERMANY)
Unless there are other special cases, I believe this information from the German health ministry applies:
Can dental and medical prescriptions for medicinal products, which were issued in the United Kingdom, still be dispensed in Germany from 1 January 2021?
According to section 2 (1a) of the Ordinance on the Prescription of Medicinal Products (Arzneimittelverschreibungsverordnung, AMVV), only medical or dental prescriptions from member states of the European Union, the European Economic Area and from Switzerland are deemed equivalent to prescriptions issued in Germany (implementation of Directive 2011/24/EU on the application of patients’ rights in cross-border healthcare). Therefore, since 1 January 2021, prescriptions from the United Kingdom may no longer be dispensed in Germany due to the withdrawal of the United Kingdom from the European Union and the expiry of the transition phase by the end of 2020.
If anyone has further official statements for the rest of the world, please let me know!
[Updated as extra information has been found and added.]