The UK will be outside the “club” and hence become a ‘third country’ for EU VAT purposes. This means that supplies from UK suppliers to customers established in the EU will follow the current rules for supplies from third countries so far as VAT in the EU is concerned. However, the place of supply of VAT may shift to where the supply is “used and enjoyed” rather than where your customer is established depending on rules in the countries concerned. For supplies of services from EU suppliers to the UK is likely to follow the rules currently in place for supplies to Norway and Switzerland.
Do pass this tip onto anybody else that might have incurred VAT in the UK.