That I would notice, I expect:
shortage of seemingly random goods in shops, empty spaces on shelves, like when Sainsburys had logistical problems about 5 yrs ago.
Worse, trying to get specialist goods will take longer (say a specific part to repair the photocopier or a bicycle), much longer waiting for arrival (extra week or so). Will be harder to source & more expensive, too.
More expensive durable goods.
Certain foods become less reliably in shops; like trying to find parsnips in July or baking apples in April.
Announcements of small manufacturing companies shutting down.
MORE obviously foreign workers, especially in NHS (Philipino & Vietnamese nurses), academia & IT (Indians). Higher salaries for the skilled workers now that UK competes more directly with likes of USA for these high skilled products of middle income countries.
Lots of consumer news stories about changes for people holidaying in EU countries (more expensive insurance, people caught out with no health insurance, overstaying visas, incorrect driving licenses, etc.)