A new variable has been introduced into the game here:
Trump retained a lawyer Wed night who showed up to the Cohen hearing today to fight elements of the warrant.
Trump and Cohen's lawyers are both fighting for the right to review the evidence before the Prosecutors do because they suspect some of it is covered by attorney-client privilege. Trump's new lawyer was able to get delay so they would have a chance to review the new filing the SDNY Attorney's office filed in Court today. They get until Sunday night to review it and prepare their response. Meanwhile SDNY is pushing back and claiming that Cohen's lawyer is trying to use attorney-client privilege as "a sword" to stop seizure of evidence. (The whole sword vs shield metaphor is one of the legal field's favorites)
It got REALLY Circus-y when Stormy Daniel's lawyer showed up and asked to be included so he could basically fight to make as many of the documents public as possible- citing the fact that some of that material invariably involved Daniels.
So this might actually work. Candidly, Cohen doesn't have a ton of leg to stand on with this. Odds are the evidence presented for the warrant was overwhelming, and there's no way this wasn't done 1000% by the book. And under normal circumstances Trump wouldn't either because his communications were actually a part of the warrant. (There's a relatively recent 6th Circuit decision regarding taint teams brought by a third party in an attorney's office raid that ruled for the third party and punched a moderately sized hole in the taint team concept. Being a third party here makes your case much stronger)
BUT when you introduce the President- suddenly this case has actual Constitutional implications. I'm guessing they'll end up appointing a magistrate Judge to oversee the taint process, but regardless of what happens the Trump legal team will appeal any ruling that falls short of what they want. It would be extremely difficult for an Appellate Court to refuse to take the case, and also very difficult for SCOTUS to refuse to take the case. So at the very least- this builds in a solid delay.