I'd say here tea drinkers are pretty varied. Drinking tea is pretty normal, we probably have it for the same occasions as Americans have their coffee, though sometimes it's coffee here to (my dad always loves his cup of coffee. I think we get peopl like you mentioned too. As for tea shops, the one I linked is pretty much set out like a normal cafe, but they have a specific menu for tea, probably not too different from a coffee shop, except not smelling of coffee or favouring a darker environment,like Costa or Starbucks. Up the road in one of the towns near me there's a tea room, the cups are dainty, floral patterned, the food mostly consists of scones and you're most likely to order a cream tea. Kind makes me think of old ladies going around the vicar's for a cream tea and scones. then there's TeaMonkey, which I go to when I visit Milton Keynes, it's a lot more modern and hip, kinda like an Internet cafe (they actually have tables with iPads on them).
However, most places you can eat serve tea, so there's not a lot of tea shops, usually coffee shops serve tea, it can sometimes taste bad (like from Starbucks) but the independent coffee shops I think do better tea (had a really nice chai in one I went to in Derby.
As for how to drink tea, for me if it doesn't have milk in it it's not tea. And it has to be hot, none of that ice tea malarkey. Some tea tastes great with an infusion of some kind, like Rooibos and cherry, but that's because I love different types of tea, the average person drinks Tetley, Yorkshire Tea, Twinings or something like that, be it black tea, Assam, English Breakfast or Earl Grey, not the fancy crap I buy from specialist tea shops.