You could also go for "continuing professional development (CPD)" in the relevant specialism".

Facharztweiterbildung refers to training as a resident while Fortbildung refers to continuing medical education (the latter does not really haver anything to do with the residency curriculum).

As the asker is located in UK, I suppose that the education will follow UK rules.

specialist der Facharzt | die Fachärztin Pl.: die Fachärzte, die Fachärztinnen
consultant (Brit.) der Facharzt | die Fachärztin Pl.: die Fachärzte, die Fachärztinnen
medical specialist der Facharzt | die Fachärztin Pl.: die Fachärzte, die Fachärztinnen
http://dict.leo.org/ende/index_de.html#/search=Facharzt&sear...

further education die Weiterbildung
continuing education die Weiterbildung
advanced training die Weiterbildung
qualification die Weiterbildung
:
http://dict.leo.org/ende/index_de.html#/search=Weiterbildung...

Domain Medical science
de
Definition nach Abschluß der Facharztausbildung durch die Ärztekammer auf Antrag erteilte Genehmigung zum Führen der Facharztbezeichnung
Term Facharztanerkennung
Reliability 3 (Reliable)
en
Term official approval as a specialist
Reliability 3 (Reliable)
http://iate.europa.eu/SearchByQuery.do?method=searchDetail&l...

Domain Medical science
de
Definition Weiterbildung des approbierten Arztes mit dem Ziel der Facharztanerkennung
Term Facharztausbildung
Reliability 3 (Reliable)
en
Term medical specialization
Reliability 3 (Reliable)
http://iate.europa.eu/SearchByQuery.do?method=searchDetail&l...

Jeder Arzt mit
abgeschlossener Facharztweiterbildung muss innerhalb von fünf Jahren 250 Fortbildungspunkte durch Teilnahme [...] an anerkannten Fortbildungsmaßnahmen erwerben.
ukaachen.de:80
Each physician with a completed medical specialist training must acquire 250 advanced training credit points by participating [...] in accredited advanced training measures within five years.
ukaachen.de:80

In Anlehnung an diesen Lehrplan hat [...] die Deutsche Gesellschaft für pädiatrische Infektiologie (DGPI) einen Lehrplan für die Facharztweiterbildung entwickelt.
bio-pro.de
The German Society for Paediatric Infectiology (DGPI) has set up a similar curriculum focusing on the training of medical specialists.
bio-pro.de
http://www.linguee.de/deutsch-englisch/uebersetzung/facharzt...

Frau Dr. med. Diana Mintzer studierte an der Johann Wolfgang Goethe-Universität Frankfurt am Main Humanmedizin, legte 1989 ihr 3. Staatsexamen ab und promovierte 1990 im Fach Augenheilkunde an der Universitätsklinik Frankfurt am Main.
Schon während ihrer Facharztweiterbildung in Darmstadt, Mainz, München und Bremen beschäftigte sie sich mit den diversen Laserverfahren am Auge (Argon, YAG, Dioden) sowie mit der Retinologie.
https://www.care-vision.de/augenarzt-laserbehandlung/dr-med-...
Dr. Mintzer studied human medicine at the J.W.v. Goethe University in Frankfurt/M. In 1989 she passed her third state exam and in 1990 she received her degree in ophthalmology from the University Hospital in Frankfurt/M.
During her advanced training as a medical specialist in Darmstadt, Mainz, Munich and Bremen, she worked with various laser eye processes (argon, YAG, diodes) as well as with retinology.
https://www.care-vision.com/Doctors-and-Branches/Our-doctors...

Die Richtlinien gelten für die von Staatsangehörigen der Mitgliedstaaten ausgeübten Tätigkeiten des angestellten wie auch des freiberuflich tätigen Arztes.
Vorgesehen ist die gegenseitige automatische Anerkennung der von den Mitgliedstaaten ausgestellten Diplome, Prüfungszeugnisse und sonstigen Befähigungsnachweise, die eine Grundausbildung (Liste siehe Artikel 3), eine allen Mitgliedstaaten gemeinsame fachärztliche Weiterbildung (Liste siehe Artikel 5) oder in einigen Mitgliedstaaten vorgesehene Facharztweiterbildung (Liste siehe Artikel 7) abschließen.
http://eur-lex.europa.eu/legal-content/DE/TXT/?uri=URISERV:l...
They apply to the activities of nationals of the Member States working in a self-employed or employed capacity.
They make provisions for the automatic recognition in each Member State of the diplomas, certificates and other evidence of formal qualifications which have been awarded to nationals of Member States by the other Member States and certify the completion of basic training (for list, see Article 3) or specialist training common to all Member States (for list, see Article 5) or some Member States (for list, see Article 7); in the last case, automatic recognition is restricted to those Member States which provide the training in question.
http://eur-lex.europa.eu/legal-content/EN/TXT/?uri=URISERV:l...

Trainees undertaking the Emergency Medicine Specialist Training Program must complete a range of assessments to become a Fellow and work as a consultant in emergency medicine.
:
Trainees complete the Fellowship Exam during Advanced Training. The exam consists of a written component and a clinical component; candidates are examined at consultant level, with a focus on …
https://www.acem.org.au/Education-Training/Specialist-Traini...

The Clinical Haematology Advanced Training Program provides specialist training in Haematology. Trainees complete their Advanced Training under the supervision of a practising Haematologist at an accredited training site.

The Clinical Haematology Advanced Training Program includes:
• 36 months (three years) of core and non-core training in a training position at an accredited training site
• Work-based assessments
• Teaching and learning requirements
https://www.racp.edu.au/trainees/advanced-training/advanced-...

Prosthodontics (Advanced Training) MClinDent
A full-time, three-year Master's programme designed for students from outside the European Union. It matches the institute's Specialty Training programme and is designed to enable students to meet training requirements in their home country.
:
All applicants must normally hold an approved dental qualification and have a minimum of two years' clinical experience post-qualification.
https://www.ucl.ac.uk/prospective-students/graduate/taught/d...

In the UK, it is a legal requirement that a doctor who wishes to practise as a substantive, fixed term or honorary consultant in the NHS must hold specialist registration. Similarly, in order to practise as a GP, a doctor must hold GP registration. A Certificate of Completion of Training (CCT) confirms that a doctor has completed an approved training programme and is eligible for entry onto the GP or Specialist Register. Between the end of the first foundation year, when doctors are fully registered with the GMC, and the granting of a CCT, there is no recognised intermediate ‘waypoint’ for doctors
http://www.gmc-uk.org/Specialties_subspecialties_and_progres...

At least one year must be advanced training (attached to a comprehensive stroke service, comprising hyperacute and acute rehabilitation units and transient ischaemic attack clinics); the greatest benefit may be achieved if this is in the later years of specialty training to build on expertise already gained in the main specialty.
http://careers.bmj.com/careers/advice/A_career_in_stroke_med...

Specialist training to CCT is through Pharmaceutical Medicine Specialty Training (PMST), a 4-year workplace training programme with assigned educational supervisors and an agreed ARCP process. https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd...

The RCP wants to ensure that trainees undertake high standards of training to ensure high quality patient care. We provide medical career information and work with our sister UK Colleges of Physicians in Edinburgh and Glasgow to set and maintain standards for physician specialist training in the UK.
https://www.rcplondon.ac.uk/medical-careers-training

Higher specialty training - 'ST3'
Following medical school, foundation training and core training, specialty training - or 'higher specialty training' - is the final part of training in the move from trainee to consultant.
http://www.st3recruitment.org.uk/about-st3

Pharma companies invest heavily in their physicians. In the UK for example they sponsor speciality training leading to a Certificate of Completion of Training (CCT), the holder of which is eligible to apply for entry to the Specialist Register of the General Medical Council (GMC) in the UK.
http://www.onlymedics.com/pharma