Will I still be able to use my insurance to drive in other EU member states after 31 December?
Yes. All UK motor insurance providers will continue to provide the legal minimum motor insurance cover for travel to European Economic Area countries (EEA - all EU countries, plus Norway, Iceland, Liechtenstein).
If you are a UK motorist, you will, therefore, not need to purchase additional third-party motor insurance policy cover when travelling to these countries with a UK-registered vehicle. You would continue to hold the same third-party cover that you do now. 
Will I still be able to use my insurance to drive in other EU member states after Brexit?
Yes. All UK motor insurance providers will continue to provide the legal minimum motor insurance cover for travel to European Economic Area countries ((EEA - all EU countries, plus Norway, Iceland, Liechtenstein).
What happens if I have not received a Green Card?
If you are not carrying a Green Card when it is required, then you will not be able to drive legally in any EU member state. If you do attempt to drive in the EU without holding a Green Card, you may be accused of driving without insurance and could be subject to a fine, having your vehicle seized or prosecution.  
The only other legal option available would be to purchase insurance locally when you arrive in the country (often known as ‘frontier insurance’). However, such insurance cover may not be widely available and may be more expensive than UK-issued policies. 
If you are not carrying a Green Card when it is required, then you will not be able to drive legally in any EU member state. If you do attempt to drive in the EU without holding a Green Card, you may be accused of driving without insurance and could face a fine, having your vehicle seized, or prosecution.  
Northern Ireland residents travelling to the Republic of Ireland:
Republic of Ireland residents travelling to Northern Ireland:
No. Valid Irish insurance discs will be accepted as proof of insurance for Irish vehicles.
Northern Ireland residents driving to the Republic of Ireland: