New legislation was passed earlier this year that will raise the requirements for the keeping of primates by private individuals in England. This will come into force on the 6th April 2026. This will cover all species of primates, other than Homo sapiens, which belongs to the order Primates.
I'm sure Wales and Scotland will follow suit eventually.
For details of the legislation see: The Animal Welfare (Primate Licences) (England) Regulations 2024