I called up for mine and the call centre operator just manually adjusted. less than 5 minutes
If it does not happen in 3 or 4 days from actual date then I suggest you call again
Do you mean at the end of the membership year or when you reach a new status tier? My experience with reaching a new tier is that it takes effect quickly (within a day or two of the required SCs/sectors posting), but the annual review is unfathomably slow.
For the last two years running I met the requirements to retain Platinum within the first 3-4 months of my membership year, but I still had to wait 2-3 weeks after my review date before they completed my status review. On the first of these occasions, I was leaving for a 3.5 week overseas trip exactly 14 days after my review date, and I wanted to ensure that I got my new card before I left, so I phoned them up. They were completely unsympathetic - they said the status review can take up to 4 weeks and can't be expedited. I pointed out that I had exceed the required number of SCs and eligible sectors over 8 months before then, so there was really nothing to review, but they came out with some waffle about everything needing to be checked to ensure there hadn't been any mistakes. I then politely informed them that I had been pooling all SCs to my wife since reaching the requalification threshold, so if they now tell me 8 months later they had made a mistake and I had fallen short, I was not going to be happy. After getting no useful response on the phone, I sent an email to complain, but as usual they just ignored it. Needless to say, I went on my trip without my new card, so it definitely took more than 2 weeks to arrive.
Last year, I had no O/S travel planned during the relevant period, so I wasn't really bothered, but as another data point, my status review was completed 13 days after my review date, and my new card arrived about a week after that. Again, this was despite having met the criteria to retain Platinum status >8 months before my review date. Yet another bizarre feature of the Velocity program, unfortunately.
To the OP - I think based on a review date of 12 June, you might receive your new card by the end of the month, but I wouldn't say that's certain.