The point of the article which you quite dramatically missed is NOT that there will be a volcanic ash cloud over the UK, it's that the economic damage caused will be just as bad as a volcanic ash cloud over the UK.
Basically what that means is that a volcanic ash cloud would cost our economy money, right? It causes issues with transport, people can't get to work, they don't produce goods. Businesses can't effectively produce their goods/services because of damaged infrastructure.
What this leads to is ON AVERAGE across everyone in the country the average person will earn less money, because there's less value being created. Say they work in a shop which sells less stuff, there's less money to pay employees. This will also go round wherein they then spend less money in other shops and so the those shops earn even less money (and so employees paid even less and so they have less money and so on... This is an economic term called the multiplier effect.
The argument the article is saying which has been shown by many economists and think tanks is that something similar will happen because of brexit... not because people can't get to work but because businesses will lose people to sell to. There will very likely be tariffs between us and the EU (basically the EU taxes any goods we send there to sell) and so our goods are more expensive. Over 45% of ALL trade that we in the UK do with other countries is with the EU! This is basically like a one shop on a street being taxed more than the other shops (because the other shops are still part of the EU) and so customers will go to the other shops. This means our companies and businesses will sell less stuff, earn less money and everyone in our country on average will earn less money.