Most Oscar Nominations for an Emmy Winner: 5
Glenn Close was nominated for five Academy Awards in the 1980s, but didn’t take home any of the major trophies until she won an Emmy for her role in Serving in Silence, a made-for-TV movie. She also won two Emmys for Damages. It's just the best example of the phenomenon of film actors, often long-nominated but without any hardware, moving to TV — whether for the additional creative freedom or, in some cases, awards bait.
Some other examples: Patricia Arquette's career was primarily in film until she starred in Medium, for which she won an Emmy — long before collecting a statue for Boyhood. And Alec Baldwin was nominated for an Academy Award in 2003, but his primary source of shiny mantlepiece hardware is 30 Rock.
And that's just in the lead categories; factor in the Guest Acting categories, and the number of big-screen actors dropping by TV shows increases quite a bit. Ray Liotta won his first major award for a guest turn on ER. Oscar nominee Laurence Fishburne (for What's Love Got to Do With It in 1993) won Emmys for HBO movie Miss Evers' Boys and an episode of the anthology drama TriBeCa. Monty Python's John Cleese has been nominated for only one Academy Award (for writing A Fish Called Wanda in 1988) while winning an Emmy for playing marriage counselor Dr. Simon Finch-Royce on Cheers. And though Gene Wilder didn't nab Oscars for acting (The Producers) or screenwriting (Young Frankenstein), he did go home with an Emmy for playing Will's boss on Will & Grace.