Limited response.
https://sg.anz.com/onlineapp/faces/optimum_rewards/redemption.jsp says:
Your unused Optimum$ will be transferred to DBS for future usage if you choose not to redeem for Cash Rebates via the ANZ Rewards Programme before 24 July 2017.
Given this, spends after 25 July cannot earn Optimum$. It is not a rolling date for conversion.
I think this because they have not said that 25 July onwards, ANZ spends will continue to earn Optimum$s, and then there will be another date in the future when remaining Optimum$ will become DBS points. It is of course possible, but I suspect it is unlikely.
Given everything, I think the new DBS card will arrive or the conversion to an existing DBS card account will happen before 25 July. But even if people don't necessarily start using the new DBS card on this date, for all practical purposes the reward earning has to be under new DBS rules.