It probably will have less of an impact in the fresh food sector - fresh meat is an expensive commodity to move such a distance and remain price competitive against UK alternatives. Labeling will be a problem but the supermarkets will find ways to re-assure the customer "All our fresh meat from Uk farms etc" doesn't contravene the rules on any individual products, whilst getting the message across.
Where it will have an impact is in the processed food sector where price will play a role as food can be transported frozen. It's not just meat, egg production is an area where things could get dangerous. The Uk egg industry supports about 10,000 jobs and pretty high standards given previous scandals. Egg supply to retail will probably not be affected for the reasons above but a huge amount of egg production goes into processed food as liquid egg. At the moment, we have EU standards so our egg producers receive a reasonable price and retain standards.
Outside the EU and with trade deals in place with say the US or, more crucially for the egg trade, Ukraine (signed this week), where a vast quantity of eggs are produced at much lower standards and a much lower price, there will be nothing to stop them hitting the uk market. UK food processors faced with tariffs on other ingredients might well be tempted to cut a cost by buying cheap eggs. Our farmers will either have to drop their standards (thus the standards in supermarkets) or more likely be undercut and put out of business. Those who do wish to maintain standards will find increased costs and competition and will either need to pass those costs on to the consumer or fold.
The lower standard eggs will enter the food chain untraceable and unlabeled. They have much higher instances of listeria and salmonella not to mention the welfare of the chickens. The only thing which might stop certain food manufacturers doing this is if they sell into the EU where they will have to declare the Ukrainian eggs and they would not be allowed to sell their products in the EU. However, there is nothing to stop this in the UK market.
I have to admit, I am struggling to have much sympathy with farmers given the number of UKIP and Brexit signs all over farmers fields. They were told this would happen but voted it through anyway (and we haven't even got to farm subsidies yet). I do reel sorry for those who voted remain and we're shouted down when they tried to raise this.
At some moment pint some hapless junior minister will have to be wheeled out to explain why there is an outbreak of salmonella and how it links to their trade policies, but don't hold your breath on them taking responsibility.
But still, blue passports eh