Marsh co-created the series with Eileen Atkins and starred in both the 1970s version of the show and its 2010 re-boot.
Atkins, who did not appear in the original series, had a role in the drama when it returned last year, but said she would not appear in a second season of the new edition, set to start taping soon and to air next spring.
The BBC said scripts for the upcoming season are being rewritten to explain the absence of Marsh's character, maid Rose Buck, while the actress recuperates.
"Unfortunately, I won't be returning to Eaton Place quite as quickly as I had hoped," Marsh said in a statement.
"I am looking forward to spending time with Rose again when I'm match-fit and I miss the cast and crew tremendously."
"Upstairs Downstairs" is about members of a wealthy family and the household staff that serves them in 1930s England.