Yes, it is permissible to immerse with a Japanese manicure, since the substances are largely absorbed into the nail. Be sure to clean the nails well prior to immersion.
There may be a slight preference for scheduling the manicure at least a few days before immersion, to allow the paste and powder to be fully absorbed into the nail, and because at that point the protective barrier is more permeable.
Please see here for a more detailed discussion of Nails and Mikveh, and let us know if we can be of further assistance.
This response was updated 15 January, 2024.