So, Camilla--
I heartily agree with Jeeves and Wooster, which I also grew up watching. Buy the DVDs-- you won't regret it.
If you enjoy Downton, you should also like Lark Rise to Candleford, a BBC series set in the late Victorian Era about a poor teenager who goes to her mother's cousin as an apprentice in the post office of the nearest, more upscale town. Among several familiar faces, Brendan Coyle left the last season of this to play Mr. Bates in Downton. It's not on Netflix, but KBYU is airing it right now, or it's available through Amazon Prime Instant Video.
Also check out the Elizabeth Gaskell mini-series collection (not actual seasons, but between them all there's at least a British-season's worth of television): Wives and Daughters (Keeley Hawes from Upstairs, Downstairs is in this one, among others), North & South (Featuring a young and incredibly hot Richard Armitage; Brendan Coyle from Downton is also in this one), Cranford, and Return to Cranford (Judy Dench, Michael Gambon, and tons and tons more). Between all of these miniseries, I think they used pretty much every British actor that looks okay in a period costume. Wives and Daughters and North & South are on Netflix, Cranford is on Amazon Prime. Return to Cranford isn't, but you can buy individual episodes.
You might also like the Hornblower series (more a series of miniseries, but still several hours of good-quality television) starring Ioan Gruffudd, about a young man who works his way up in the British Navy during the Napoleonic Wars. More action-oriented than others, but still highly enjoyable (and full of cute British boys). Not available on Netflix or Amazon Prime, unfortunately.
My family just started watching Garrow's Law, a historical drama based on the story of the real-life solicitor who pretty much single-handedly reformed the British justice system, and it's very good. We had to buy the DVDs, though. It's not on Netflix or Amazon Prime, unfortunately. You might also like Bramwell, starring Jemma Redgrave, as a female doctor in the Victorian era. I've seen the first few episodes and enjoyed them, although I hear the quality declines somewhat over the series. That one is on Netflix.
If you like mysteries, that opens a whole new level of period mystery series-- everything from the Jeremy Brett Sherlock Holmes to the long-running Poirot starring David Suchet to Marple (The more recent series is the one I'm thinking of, which includes guest appearances from everyone from Benedict Cumberbatch to Dan Stevens) to Foyle's War and beyond. At least some episodes of all of those are on Netflix.
And I can't help but recommend a few movies you might not have seen, as they seem to have gone under the radar here in the States-- The Winslow Boy, starring Rebecca Pidgeon, Nigel Hawthorn, and Jeremy Northam, is a beautifully-understated drama about a father's fight to clear his young son's name; the adaptation of Oscar Wilde's An Ideal Husband, starring Rupert Everett, Jeremy Northam, Minnie Driver, and Cate Blanchett, is gorgeous and hilarious (features some extremely brief, out-of-focus nudity at the very very beginning of the show, but blink and you'll miss it); The Importance of Being Earnest, starring Rupert Everett, Colin Firth, Reese Witherspoon, and Frances O'Connor, is likewise hilarious and features Colin and Rupert singing a serenade; and Amazing Grace starring Ioan Gruffudd and Benedict Cumberbatch, among others, is based on a true story about the fight to end the slave trade in Britain and British territories.
Happy watching.
