They won't because our domestic legislation is ours. I doubt it would be a good political move for any party to suggest that we removed the origin of our own produce, especially within our own domestic market. Therefore if we had a trade deal with the US or anyone for that matter, if we agreed we would import their produce without the need to display it's origin, it would still be a case of UK consumers having a choice between British labelled produce and 'Other'. If you want to avoid 'Other' you can without needing to know where 'Other' relates to because you will have chosen to buy the British labelled produce instead.
What you're suggesting is they'd force the UK to for instance bottle scotch whiskey as being from some anon source. That wouldn't happen because US consumers would want to be able to identify such goods and those selling them in the US would want that too as they attract a premium price compared to a US made product.