I think you might be confusing time signature (roughly, how many beats per measure) with tempo (how fast the beats are).
Aha - THIS is the version of the M:I theme I was remembering. It starts in 5/4 before segueing into 4/4:
In 1996, two members of U2, Adam Clayton and Larry Mullen Jr., remade the Mission Impossible theme track. They made two versions of the song: the main one and another called “Mission Accomplished.” The main version played during the end credits. Unlike the original, most of this version follows a regular beat, except for the beginning.
It’s not a cover version, but (British) TV show The Bill changed it’s theme tune from 7/4 time to 4/4 time. Thus:
j
I knew this would come back to haunt me!
Solsbury Hill by Peter Gabriel is in 7/4 time.
The synth-pop cover by Erasure is in 4/4, except for one bar of the chorus that reverts to 7/4.
- NOT Salisbury, by the way. Different place entirely.
Lady Gaga sung the national anthem at Biden’s inauguration in 4/4 (mostly-ish). It’s normally played in 3/4.
(too late to edit)
That should be sang, not sung.
(never mind)
Many (most?) covers of “The Impossible Dream” change it from 9/8 to 12/8. Elvis’s does. Sinatra’s doesn’t.
“Memory (from “Cats”)” has one measure of 10/8 in the middle of the verse somewhere — the rest of the song is 12/8 — and everyone except Andrew Lloyd Webber (Streisand, Manilow) makes that measure 12/8.
It may be my imagination, but most renditions nowadays seem to go briefly into 2/4 time at the end of the first verse (“TWI-light’s LAST-glea MING”). In older recordings, you can hear a clearer 3/4 stress pattern (“TWI-light’s-last GLEAM-ing”).
Novuelle Vague’s cover of Killing Moon is in 3/4 time, whereas the original is in 4/4 (at least the snare is).
The reason I started off insisting that it’s in 3/4 is that I’m so used to this syncopation:
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12
…which you can indeed write as 3/4 time. It’s got one more couplet at the end that Mission: Impossible doesn’t have.
Still in 4/4, like the original.

The reason I started off insisting that it’s in 3/4 is that I’m so used to this syncopation:
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12
Ha! Take 5 and America are close cousins indeed!
There was also the Limp Bizkit theme to Mission Impossible 2, “Take a Look Around,” that recontextualizes the theme in 4/4.
And that 3+3+2+2+2 beat is an example of something called “hemiola” which basically gives a feeling of alternating between duple and triple meter. It can be thought of as alternating 6/8 and 3/4.