It appears that the options I will describe below are temporary and linked to the Dressing Downton exhibit at the Lightner Museum, but I hope either site changes its mind and starts offering afternoon tea on a regular basis. Why not?
My husband and I recently spent half a week in St. Augustine, one of our favorite places to visit in the southeast. This time around we decided to do more of the historical places, including the Lightner Museum, the Castillo de San Marcos, the Flagler history tour, a carriage ride, and even a tour at the place we stayed, The Collector Inn. Since the Dressing Downton exhibit was also at the Lightner, they offered afternoon tea at the Cafe Alcazar (now through January 4, 2018.)
The tea required reservations, and we were very fortunate that someone working there took pity on us and added a table, as they were technically booked when we tried making reservations. The cafe is in what used to be the pool of Flagler's second huge hotel in St. Augustine, and is now the center of the Lightner Museum. Tea started with a soup or salad course along with iced tea, while they took our hot tea orders (pretty smart, I thought!). The rest of the tea is portrayed above, with sandwiches on the bottom, scones and a cookie in the middle, and tiny sweets on the top. While we had tea, they played music from Downton Abbey. They sent us home with samples of the hot teas on offer, so everyone was able to try at least two kinds - one in the pot, and one in loose leaf brewed at home.
It looks like after this particular exhibit moves on, and afternoon tea goes along with it, you can continue to lunch at the Cafe Alcazar.
I recently discovered that you can also have afternoon tea and view Gosford Park at the nearby Corazon Cinema and Cafe. That's a place we will want to visit in the future, because we didn't make it this time around.