There's the Vintage Tea Room on Colomberie in town, but there's other
places to have a nice afternoon tea that aren't actually Tea Rooms, including many of the upmarket hotels.
Chateau la Chaire is actually a bit out of St. Helier, it's at Rozel in St. Martin. If you're wanting something in town (or close to it) Longueville Manor would probably be my first choice, but you've also got the Pomme d'Or and Grand Hotel. In St. Brelade there's the Atlantic and L'Horizon. I'm sure there'll be many more too.
Other places will have a great selection of cakes and sandwiches, so although maybe not the traditional 'Afternoon tea' you can still have something tasty!
jdtaylor: The problem with the banks is that some of them are offshore divisions of UK companies, so not technically the same bank. Confusing, I know! ATMs use the link system so no charges there. There may be a charge for using Debit or Credit cards at ATMs, but there's no charge for in-store purchases on a UK card.