Brexit takes place on 31 January, and the UK will no longer be member of the European Union. Many arrangements will stay in place for a transition period whilst a deal between the UK and the EU is negotiated.
The current advice from the UK government is that during the transition period (until 1 January 2021) arrangements for driving in EU countries remains unchanged.
At the end of the transition period you will need to get yourself an International Driving Permit and a green card to ensure your insurance is valid. You will also need a GB sticker for your car.
Note that Brexit does not affect gap insurance. Your gap insurance policy will still be valid if you drive in Europe - however you do need to be covered by a fully comprehensive cover for the policy to be valid in the first place and this will require a green card from your insurer.
https://www.gov.uk/visit-europe-1-january-2021
Where do I get an International Driving Permit?
International Driving Permits can be obtained from the Post Office. You will need a passport photo and your driving licence. Currently different EU member states require different length permits for non-EU drivers - for instance Spain requires a one year permit and Germany a three year. If you are touring the EU you should get both to be on the safe side.
How much do International Driving Permits cost?
The International Driving Permit costs £5.50 pre document (you may need two if you are driving between countries with different requirements).
How do I get a green card?
You will need to call your insurer before you set off, let them know the exact dates that you require it for. And (naturally) they will charge you.
Whilst the government advice is to allow a month to receive the green card, our advice from our insurer is that you don’t actually need the bit of paper to travel, they’ll have a note of it if anything unfortunate happens. It is currently not clear whether you will need to carry your green card for examination by the authorities