| Year | Award | Category | Work | Result | Ref. |
| 2021 | Critics' Choice Television Awards | Best Actress in a Drama Series | The Crown | Won | |
| 2021 | Golden Globe Awards | Best Actress – Television Series Drama | Won | |
| 2021 | Broadcasting Press Guild | Breakthrough Award | Won | |
| 2021 | Hollywood Critics Association TV Awards | Best Actress in a Streaming Series, Drama | Won | |
| 2021 | Online Film & Television Association | Best Actress in a Drama Series | Nominated | |
| 2021 | TV Times Awards | Favourite Newcomer | Nominated | |
| 2021 | MTV Movie & TV Awards | Best Performance in a Show | Nominated | |
| 2021 | Satellite Awards | Best Supporting Actress – Series, Miniseries or Television Film | Nominated | |
| 2021 | Screen Actors Guild Awards | Outstanding Performance by a Female Actor in a Drama Series | Nominated | |
| Outstanding Performance by an Ensemble in a Drama Series | Won | |
| 2021 | Primetime Emmy Awards | Outstanding Lead Actress in a Drama Series | Nominated | |
| 2022 | Laurence Olivier Awards | Best Actress | Anna X | Nominated | |
| 2022 | Diva Awards | Actor of the Year | Nominated | |
| 2022 | The Queerties | Closet Door Bustdown | Nominated | |
| 2022 | Toronto International Film Festival | Tribute Award for Performance | My Policeman | Won | |
| 2024 | Independent Spirit Awards | Best Lead Performance in a New Scripted Series | A Murder at the End of the World | Nominated | |
| 2025 | Saturn Awards | Best Supporting Actress | Deadpool & Wolverine | Nominated | |
| 2025 | Critics' Choice Super Awards | Best Actress in a Superhero Movie | Nominated | |
| Best Villain in a Movie | Nominated |