Porco et al studied a situation where a phlebotomist in California admitted to intentionally reusing needles.
They estimated likelihood of HIV transmission (as well as hepatitis B/C) based on population prevalence (the first person needs to be HIV+), transmission probability (enough virus needs to remain alive and get transferred to the second person to cause infection), and rate of needle reuse.
For HIV, the risk ranged from 1.4 in 100 million in the best case scenario (low prevalence, low transmission rate, correct number of reuses reported) to 6.8 in 1 million (more people already HIV positive than expected, highest transmission rate, 100X more reuse than reported). Even the best case scenario described is a situation where every single needle was being reused 5-10 times.
Even this best-case scenario is much worse than an event of accidental reuse, because this is all based on a scenario of known, intentional reuse of every single needle.
In summary, the risk of HIV transmission from accidental reuse is effectively zero. There is no plausible way in the work flow for a used needle to be confused with an unused one. You are only at risk in a situation when needles are intentionally reused, and even then the risk is low unless you are in an area where HIV prevalence is very high. Intentional misuse has occurred even in places like the US (and perhaps Europe; I am aware of US cases but not European) but is more likely in places with stronger incentives to save costs, for example a recent event reported in Pakistan.
@CareyGregory had a good suggestion in a comment that if you are at all concerned, you ask the staff to walk you through the procedure: if nothing else, this should reassure you of the process.
Porco, T. C., Aragon, T. J., Fernyak, S. E., Cody, S. H., Vugia, D. J., Katz, M. H., & Bangsberg, D. R. (2001). Risk of infection from needle reuse at a phlebotomy center. American journal of public health, 91(4), 636.