'Tacky' not a saying - a 'saying' is an expression, like a proverb or aphorism. 'Tacky' is just a word! But, to answer your question: yes, it is used in both British and American English. In its literal sense, it means 'slighty sticky', as in "The glue is still tacky. You need to wait another few hours until it's completely dry". Colloquially, the word is used to mean lacking in taste. The Trump family's sense of interior decoration, for example.

I'm not sure about the U.K. either, but it's pretty common to throw a tacky Christmas sweater party in the U.S. hahaa! The guest who wears the ugliest sweater wins the prize! :)

Don't know about the U.K. but it's very common in the US.