I've recently rewatched the entire series, so now I'm pretty sure about most of them.
Here is what I think:
Robert Crawley: ISFJ
Cora Crawley: ESFJ (uses more Ne than her husband)
Mary Crawley: ENTJ definitely
Edith Crawley: ISFJ
Sybil Crawley: INFP definitely
Violet Crawley: ESTJ (So much sass! xDD She is so entertaining.)
Isobel Crawley: ENFJ
Matthew Crawley: INFP
Rose MacClare: ESFP obviously
Tom Branson: ENTP
Thomas Barrow: INTJ So much... (also enneagram 3w4)
Sarah O'Brien: INTJ (Unsure, might be xSTJ)
Phyllis Baxter: INFJ
Joseph Molesley: ISFJ
Daisy Robinson/Mason: ISFP (She has a very very strong Fi and she often acts without thinking which she later regrets.)
Jimmy Kent: ESFP
Ethel Parks: ENFP (or ESFP, but to me she seemed idealistic, dreamy and outgoing in an ENFP sort of fashion, not necessarily like a Se-dom.)
Anna Smith/Bates: ISFJ (or INFP, but definitely IxFx and she uses Si/Ne)
John Bates: ISFJ (or ISTJ with well-developed tertiary Fi, not sure)
Charles Carson:ISTJ
Elsie Hughes: ISTJ (A kinder, less rigid version of the type than Carson, but it's good tertiary Fi what she uses, not Fe imo.)
Mrs. Patmore: ESFJ What's her firstname? I don't think they ever mentioned it. :O (Anyway, she is not a Se-dom, like many have typed her, she is totally a traditionalist and has trouble accepting all the changes around her. Many times the other characters even joke about this.)
Gwen Dawson: INFP (We don't know too much about her, but this was my impression.)
Dr. Richard Clarkson: ISTJ (unsure...)
Andy Parker: ISFP
Septimus Spratt: ESTJ so funny...
Gladys Denker: ESFJ so horrible...
William Mason: ISFJ
Lavinia Swire: ISFJ
Rosamund Painswick: ESFJ (not sure though)
Alfred Nugent: ISTJ maybe
Ivy Stuart: ESFP
Henry Talbot: ESFP (or ESTP)
Bertie Pelham: ISFJ I guess, but really not sure...
Richard Carlisle: ENTJ
Some very minor characters, we don't know much about them, so mostly just guesses:
Evelyn Napier: ISTJ
Martha Levinson: ENTJ?
Harold Levinson: ESTP?
Sir Anthony Strallan: ISFJ
Sarah Bunting: ENTJ
Anthony Foyle/Gillingham: xSTJ?
Horace Bryant: ESTJ
Michael Gregson: ISxJ
Prince Kuragin: ISTJ?
Edna Braithwaite: ENFJ?
Charles Blake: INTJ?
Mabel Lane Fox: ESTJ
Atticus Aldridge: ESFP
Jack Ross: ESFP
Richard Grey/Lord Merton: xNFx??