A few additions to Organic Marble's answer.
The plan detailed in the CAIB report shows how difficult it would have been to mount a rescue mission.
Even though Atlantis was being prepared for the next mission, they would have had to cut mission preparation time in half. No time for tests and checks. Everyone working frantically under gigantic pressure with no room for mistakes.
Meanwhile Columbia's crew would need to extend the mission far beyond the designed on-orbit lifetime of the Shuttle, with air (actually carbon dioxide scrubber canisters) being the most critical limitation.
"Accelerate" is a prosaic word for the herculean effort that would have been needed. Activities that normally take place across weeks or months would have to happen in hours or days. Civil servants and contractors at KSC would have to begin 24/7 shift work, keeping the lights on and the process running every hour of every day, for a minimum of 21 days, to power Atlantis through checkout and make it ready to launch.
Three unceasing, brutal weeks of 24/7 shift work—and that's with absolutely no margin factored in for errors or failures. The Orbital Processing Facility team, the Vehicle Assembly Building team, and the Launch Complex 39 pad team would have had to get every one of the millions of steps right, and every component of Atlantis would have had to function perfectly the very first time, or it would all be wasted.
From the CAIB:
It should be noted that although each of the individual elements could be completed in a best-case scenario to allow a rescue mission to be attempted, the total risk of shortening training and preparation time is higher than the individual elements.
A spacewalk to assess Columbia's damage would have been difficult. Columbia didn't have SAFER jet packs on board, so its crew would have had to free-climb across the outside of the shuttle (a precarious affair without handholds) to get to the wing.
From the Ars Technica article:
The error-free rescue of Columbia would have depended not just on the flawless execution of teams at all of the NASA centers but also on an unknown number of events that happened days, weeks, months, or even years in the past leading up to the mission.