In my experience with both the Delta Platinum and the Delta Reserve Business, the sign-up bonus posts a few days after hitting the min spend (after last transaction goes past pending to posted). Not sure if it’s a fixed amount of time or certain day(s) of the week when it checks to see if you’ve hit it.
The Miles Boost/Status Boost thing is different and becomes available at the redemption link shortly after the statement closes and the PDF has been generated (after the min spend has been hit). In my case the statement close date was the 27th, but the PDF and statement ready email came the 28th at 7pm. I checked the link shortly after that and was able to redeem the boost miles. I also got the email from Delta with the link a couple of hours later.