None of these alleged facts are true.
Monkeypox was first identified when shipments of monkeys from Singapore to a Copenhagen lab were observed to present symptoms of a pox-like disease in 1958 - prior to any of the monkeys being used in any trials.
Initial and subsequent investigations (late 1950s through the 1960s) demonstrated these (and other, later) monkeys were infected with a virus that resembled - but was different from - both the
variola (smallpox disease) and
vaccinia (smallpox vaccine) viruses. The new virus was determined to be closer to
variola than
vaccinia. Decades later, after more sophisticated genetic testing, it is believed the monkeypox and
variola viruses are cousin strains, both descended from cowpox. There is no basis for believing monkeypox represents a derivative from the
vaccinia virus - i.e., any smallpox vaccine being used from the late 1950s onward.
As early as the initial investigations it was demonstrated that inoculating monkeys with smallpox vaccine (i.e.,
vaccinia) prevented the presentation of any visible monkeypox symptoms. This alone rules out the claim the first monkeypox-infected monkeys presented with infections
after being involved in smallpox vaccination trials.
The first confirmed human monkeypox infection involved a child in 1970. There's no basis for claiming humans hadn't been infected with monkeypox until that time. Owing to the close similarity between the monkeypox and
variola viruses, as well as the limited diagnostic resolution of serological testing of that era, there's a high probability that any earlier monkeypox infections in humans were simply attributed to regular smallpox.
Monkeypox isn't limited to monkeys. Some monkey species don't even present symptoms when inoculated with the monkeypox virus. The most widespread populations of wild species known to exhibit traces of exposure to, and to harbor, the monkeypox virus are rodents rather than monkeys.
See, for example:
Monkeypox Virus (a 1973 summary of everything learned over the preceding 1.5 decades since first discovery)
https://journals.asm.org/doi/epdf/10.1128/br.37.1.1-18.1973
A review of experimental and natural infections of animals with monkeypox virus between 1958 and 2012
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3635111/