I don't really have absolute pitch, but I have a decent sense of relative pitch, which often helps me to link chords/notes to whatever key I'm in without actually checking each one note-for-note (I still double-check of course but this saves tons of time on a first draft).
For melodies, I usually transcribe them by comparing intervals between the notes to save time, though sometimes in more complicated or accidental-heavy melodies I check them note-for-note. For chords, I can usually tell what chord something is by listening to it, but sometimes, like in a big orchestra piece for example, they sound a little muddy or muffled, so I check those more thoroughly. Rhythms I often get just by listening, but sometimes in slow pieces or oddly-timed pieces where the rhythm is unclear, I slow the song down and subdivide the beats to find rhythms more accurately. For finding a time signature, I listen to entire song as a whole and go based off of the rhythm and melody usually. And finally, for key signatures I usually just try to find the tonic, but this can be difficult for some songs; in those I try to go off of the chord progression and choose the most logical fit for that.
When I actually start transcribing a song, I'm usually all over the place lol. Sometimes I just do a melody to start, sometimes just the bassline, sometimes just an entire LH part, or sometimes just everything at once while working measure-by-measure. I don't really have a structure, but I get the job done either way I guess.
Getting good at figuring out all these things quickly and efficiently definitely comes with practice. Chord and note identification exercises online for example are really helpful. Again though, a lot of the stuff I do when I transcribe isn't necessary to transcribe really, but saves a lot of time when doing so. But if someone's new to transcribing, I really recommend checking songs note-by-note and just being as thorough as possible. That's how I started.
Oh, and I guess I should include what I use to transcribe, that'd probably be helpful lol. I use my keyboard, an online keyboard if I'm too lazy to go across my room to use my actual keyboard, Audiostretch, Audacity, and of course a metronome to find the tempo.
That was a mouthful lol, but yeah. That's what I do in a nutshell.