You actually have until the first pre-season game to get your salary cap fixed, so you have plenty of time to get your financial house in order. You will see the warning banner until it's fixed, but the AI won't make those cuts until the first pre-season game and then only if you are still over the salary cap.
The other part of your question which bears mentioning, if your next stage
was the first pre-season game and you were at a risk of the AI cutting your highest-paid player to get you under the cap, any cuts you have set up will take place first - so if that were to bring you under the cap then the AI will do nothing. The AI only makes those cut(s) if, after processing all that you have queued, you still violate the salary cap or the roster limit on a stage that it checks (game stage for both, training camp for roster limit only).