There are a number of different possible ways of measuring the threshold of tempo perception. It's one type of task to say to someone: "Play me a tempo of 120bpm" out of thin air with no context, and then to measure how accurate they are. It's a completely different task to say: "We'll start you off playing at Tempo A.
Maintain a consistent tempo for a minute, and we'll measure you." It's yet another task to say: "Here's an excerpt at tempo A. Now here's a short excerpt at tempo B. Are they different (faster/slower) or the same?" And it's even potentially different to say: "Here's a piece at tempo A. Tell me whether this gradually speeds up or slows down or stays at a constant speed."
All of these kinds of things (and more) have been studied in music perception studies. There are somewhat conflicting results, but here are a few useful datapoints:
- When subjects (including non-musicians) are asked to reproduce songs they know well, most are able to perform a tempo within +/-8% of the actual tempo of the source recording, and around half are able to get within +/-4% (which would be roughly a range of 115-125bpm for a nominal 120bpm tempo).
- When asked to continue tapping a tempo with a given beat, the range of accuracy was measured at 7-11% and within 3-4% accuracy when subjects tap along with a continuing beat.
- Another type of task looks at otherwise steady rhythmic pattern of beats with one beat (or more) potentially "off" by a bit. Depending on the task, accuracy can be a bit better, averaging around a 3% displacement where a beat is noticed as "out of tempo," with sensitivity going down to around 2% or lower.
- An empirical study of jazz recordings found tempo during performances of groups stayed within 5% variation even when involving significant improvisation. Accuracy between takes or when resuming a previous tempo showed a similar level of accuracy. However, these are actual recordings where presumably the highest priority was not always tempo accuracy.
- Finally, there's a question of ability to detect tempo "drift" brought up in the question. Some studies have shown that it takes a difference of around 6-8% for listeners to detect a clear alteration in tempo, in agreement with some of the other data above. However, when a different paradigm for testing tempo shifts was used to allow listeners to gradually move toward increasingly small changes around a given tempo, the "just noticeable difference" was measured to be significantly smaller, on the order of 0.27% on average for the subjects in that study. Note in that last study, however, that most subjects have some sort of "internal drift" in over the course of the study in their continuous changes in tempo, so total discrimination capability in the best of circumstances was probably on the order of within 1% or so, or within the the 1 bpm threshold for reasonable tempos that OP asked about.
- This last finding accords with measurements of professional drummers asked to maintain tempos for 20 seconds, showing that on average they could maintain accuracy within a tempo drift of within 0.2%, though some variance up to around 1% was observed. However, it's important to note that these were only measured for internal consistency -- some of these drummers, for whatever reason, began at a significantly different tempo from the reference tempo they were provided by a metronome (in one case, close to 10% off).
These last couple findings perhaps come closest to answer the OP's question. In ideal circumstances, a trained drummer can probably maintain a tempo to within 1% accuracy, which would be within about 1 bpm for reference tempos around the mid-range of 100bpm. (However, note that's maintaining a starting tempo -- which may not even be the intended starting tempo given from a metronome, just the one the drummer "feels" and begins. And note that in real-world circumstances, drummers are not typically just tasked with "Maintain accuracy in tempo, no matter what.") And under ideal circumstances, listeners may be able to detect differences of tempo on that order too, when they are juxtaposed against each other, and listeners are allowed a long time to practice on listening to that specific tempo.
But such ideal circumstances are unlikely to take into account real-world performance conditions, where the range of somewhere around +/-5% is a more reasonable estimate of when most people (both musicians and non-musicians) begin to feel a distinctly different tempo, as shown by numerous studies.
On a slightly unrelated note, to respond to another point brought up in the question about 440 Hz vs. 439 Hz, it's important to note that human ability to discriminate difference in pitch is actually very fine in ideal circumstances and with certain tasks. Certainly OP is correct that there are very few (if any) humans capable of distinguishing 440 Hz vs. 439 Hz without a reference pitch, using only absolute pitch capabilities. But certainly most professional musicians would be quite capable of noting a difference of 440 Hz vs. 439 Hz when comparing pitches during tuning of instruments, as that's an error about 4 cents, well within the frequency discrimination abilities of professional musicians. (Even if comparing only sine waves, those two pitches would have a noticeable beat frequency, which musicians would notice during tuning and try to correct.)
However, if asked to compare those two frequencies outside of extended tuning against each other, almost no one would be able to hear a difference. Again, as in tempo perception, the nature of the specific task matters.