I've got GNOME set up in English, but my region to be the Netherlands. It will helpfully display local dates, but that also results in the month names being in Dutch.
Unfortunately I don't really know what LC_TIME is :) But if I look through the calendar widget that drops down when I click the time in the top bar, is says e.g. that the previous month is called "augustus" and that today is "vrijdag". Looking into my "Region & Language" settings, my language is set to English (United Kingdom), and formats is set to Nederland (Nederlands), which is said to cover numbers, dates and currencies. Especially for currencies I'd prefer my local currency, but I'd prefer for numbers to use the UK system, and not quite sure what I'd like for dates as long as it's not the US system.
If by "local dates" dates you mean the format rather than timezone then LC_TIME=en_DK will give you English text with RFC3339 YYYY-MM-DD formatting.