Not sure apply the waiting period of dependent being added to MSP. But new application to MSP generally have a 3 months (current month + 2 months) waiting period.
Best is to contact MSP to confirm.
The site to which
@Ponga referred also says:
When Does Coverage Begin?
New (and returning) residents are required to complete a wait period consisting of the balance of the month in which residence in British Columbia is established, plus two months before benefits can begin. The date residency is considered to have been established is determined by the Ministry of Health based on a number of factors, including the type of immigration status held.
For spouses and children who apply for Permanent Resident Status while in B.C., the wait period begins on the date the sponsorship fee (if applicable), application for permanent residence fee and right of Permanent Residence fee have been paid in full to IRCC. If all fees are not paid at the same time, the wait period begins on the date the last fee is paid.
Here, the OP says he intends to start the PR application process in June and wants to know if wife can be added "once applied". It looks like "yes", if the fees are paid in full when applying.
My wife arrived in BC in December. In January we walked into a BC Services office, showed her PR card, filled out a bit of paperwork, and her health card arrived in the mail about 10 days later.