As you are moving on January 2020 and up to January 2021 nothing is expected to affect you during that period as UK will still be in the so called transition period after leaving the EU on 29th of March 2019.
Of course the above statement is closely related to if the UK will leave the EU with a “deal” or “no deal”. And it is valid in the event of a “deal”.
In the event of Brexit with a “no deal” between UK and the EU we will provide you further information as these are available.
We would like to let you know that our team has “champions” (as we call our Brexit experts) in Greece who will be happy to assist you furthermore.
Since you are moving for work indefinitely you will need to follow the normal procedure to acquire all the legal documents in the new country for you to be eligible to work.
These documents include the equivalent of VAT (called AFM) and the equivalent of NIN (called AMKA) in Greece.
For more in details information we can get you in touch with one of our greek champions.
You could also have a look on the below official UK Government links:
Please let us know if we could help you any further.