In the case of cover songs, it's pretty easy. But, it really depends on how well you know the song, and how disciplined you are when it's time to recall it. In my case, at age 56 and with a heavy background in '50-80s music, the oldies, classic & new wave hits are embedded in my soul (along with most album tracks by certain fave artists - try me on Beatles or Elvis Costello anytime!) I can recall parts, arrangements, and tempos on demand. But that's not enough. It's how I do it that helps with accuracy.
I don't hum or sing or tap my feet, because that's creating the song/tempo, and I'm probably affecting the song with various personal factors such as how tired I am, whether I like or dislike the song, etc. Instead, and this takes a somewhat conceptual approach... I listen to the song playback in my head. I hear it as if I was listening to a CD of the radio. No tapping, humming, grunting, or any other personal input from me, and it just takes a few bars of the most familar part of the song to me (usually the hook.) I hear it, and it's correct to within a few BPM, and usually in the correct key! With that same recall, I can correctly play a song I know, fills and all, even if I've never played it before. It just plays in my head while I play along to it.
Knowing a song and its tempo also helps call out tempos by #. When someone wants a song at 120bpm, I can nail it by hearing a known 120bpm song in my head, such as Dim All The Lights by Donna Summer. Need 98? I hear Eminence Front by The Who. 80bpm is We Will Rock You by Queen. Well, all are close enough. In my case, I know certain tempos because they're drilled into my head night after night, year after year on the road, and my hands and feet know those tempos.
Admittedly, all of this depends on being absolutely familiar with an existing song to recall it properly.
In the case of originals, it's just a matter of checking tempos with a metronome (for consistency) and getting the feel for those songs. Once they're burned into your heart and soul and hands & feet, you'll be able to recall them with just a few seconds of "playback" in your head. Or, and this is common, use a metronome to set the tempo so you can count it off the same each time. You don't have to play to the click, just get your countoff from there, and nobody can ever accuse you of starting too fast or slow. Of course, maintaining the tempo is the other half of the equation!
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