I think it's a beautiful song, but it doesn't seem to stick to a key
signature very well.
I'm not sure where that happens. Here's some sheet music, and the whole thing stays in a single minor key.
a) Would you consider this song atonal?
No. It's one of the most tonal uses of a minor key in popular music I've actually seen recently. It's pretty much entirely built on a i-VI-iv-(V/V)-V cycle in its chorus, which is a standard pop sequence and a standard chord progression that goes back to roughly when tonality emerged as a thing about 300-400 years ago. The verses make use a variant i-VI-IV-iv7 cycle that's built around the standard Bond theme (in this case, effectively a 5-♭6-♮6-♭6-5 repeating melodic cycle), but which gives way to the standard cadential resolutions as the chorus does.
b) If so, what makes it so enjoyable?
I'm not sure how to answer this. Does this question assume that atonal music can't be enjoyable? Why? Strands of atonal or sort of "wondering tonality" occur in a lot of movie music, though this isn't really an example of either.
(To be clear, my answer isn't trying to be argumentative. I'm just a bit confused by the question.)
EDIT: On re-reading the question, perhaps the confusion is about the frequent occurrence of raised 6th and 7th scale degrees. (In the sheet music I linked in C minor, these are the A-naturals and B-naturals.) These chromatic notes are actually very standard in the classical minor mode and really part of what it means to be "in minor." However, many modern pop songs tend to be more modal and don't necessarily use them as much, so maybe it seems odd to some people to see them so frequently. But they're not notes that are "out of the key" -- the B-natural is the leading tone and standard in minor to resolve to the tonic note C. The A-natural is mostly used (as I noted) to invoke the "Bond theme" in one of the voices, but it is also a standard occurrence in minor keys.
The only other accidental I believe is the raised 4th scale degree (F-sharp in the sheet music), which is used to create a leading tone to the dominant note. In harmonic terms, this is known as a secondary dominant or applied dominant. And this is the most common way to use such a chord in a key.
These notes may seem to be "outside" the key signature, but they're actually the most common chromatic notes using in a (tonal) minor key. "Atonality," on the other hand, implies that the music doesn't have a standard note/chord that functions as the "center." As the music keeps returning repeatedly to C minor and G major (the dominant of C minor) in the sheet music, it's clear that C minor is the tonal center.