Free trade with the US requries harmonisation of standards, which will then cause problems with trading with the EU.
Producers will not be obliged to lower their standards, but they could if they wanted to.
Lower standards are often cheaper, and UK producers will be competing with cheaply produced US food, so the temptation will be there.
To echo a pp poster, you can choose to shop at Waitrose for good quality meat, but the lower quality stuff will turn up everywhere from schools, takeaways, fast food restaurants etc.
This will disproportionately affect the poor, who may not want to eat such food, but may not be able to afford shopping at Waitrose. Poorer areas are much more dependent on their local shops as people can't afford to travel elsewhere. So it will be a question of what those shops choose to stock.