The UK could easily feed itself at home today, granted we would not have the choice of produce or many tropical foods that we have today, basic food, it could be done.
That is not the choice we face today, do we want food produced to UK standards or to lower standards from overseas. After WW1 farming went into steep decline with the cheapest option from across the globe imported, nobody considered welfare, or growth promoters or the environment. Much farmland went out of production almost to a subsistence level, then came WW2 the UK had no farming infrastructure, only with massive imports of machinery was it able to recover.
The CAP is designed to ensure Europe is self sufficient in most commodities, it has been very successful, not only can we feed ourselves but we can improve quality and environmental standards as well. Those higher standards have to be paid for and the cost is passed on in higher prices, even so food has never been cheaper, the average worker has to work less hours to feed the family than ever before.
This has been achieved by farms being much more efficient, larger units selling direct to the supermarkets, those retailers becoming closely involved in the production methods and standards. You can compare it with “just in time” you hear about in factory production.
If UK production is reduced some foods could easily be sourced overseas, Wheat, Chilled meat are two examples, tropical fruit would not change, it is fresh food that would cause problems. If it is not grown here Europe is the only alternative, so vegetables, potatoes and salads would be grown in Eastern Europe or the Mediterranean.
We will again be dependent on others to feed the nation, maybe you don’t care about that as long as it is cheap. If that is the case should hope that the EU stays together, if it splits up into countries or regions that are fighting each other for advantage, the risk of conflict and supply disruption increases.
On one hand Brexit is about independence on the other we are becoming more dependent on others for supplies of all sorts.