There is European commission regulation
No 805/2011 of 10 August 2011
laying down detailed rules for air traffic controllers’ licences and certain certificates pursuant to Regulation (EC) No 216/2008 of the European Parliament and of the Council
And it says that member states shall recognize both ATC and medical certificates issued by other state
Article 29
Recognition of licences
1. Member States shall recognise air traffic controller and student air traffic controller licences and their associated ratings, rating endorsements and language endorsements as well as associated medical certificates issued by other Member States in accordance with this Regulation.
So I would say that answer is YES. However I think not all States follow it. Let's take adverts on FG,when it says applicant shall have valid
UK issued licence. Which in my opinion is discriminating for applicants from abroad,who hold other member states issued licences,thus their licence shall be recognized. So only Unit training would be required and nothing else.
Regards.