The SFS novelization by Vonda McIntyre spends several chapters filling in the interval immediately after TWOK before eventually starting to adapt the movie proper, and it posits a gap of only about 3 days between movies, which seems way too brief. I think it's more likely a matter of weeks, however long it takes to limp back to starbase for repairs and so on.
The novel The Genesis Wave Book One by John Vornholt contains a chapter written by David Mack that's an official Starfleet report on the Genesis affair, and it gives precise dates: The Genesis planet was created on March 26, 2285 (4 days after Kirk's birthday, which is the same as Shatner's), and destroyed on April 29, 2285. That's about a month between movies, which seems about right to me.
The novel The Genesis Wave Book One by John Vornholt contains a chapter written by David Mack that's an official Starfleet report on the Genesis affair, and it gives precise dates: The Genesis planet was created on March 26, 2285 (4 days after Kirk's birthday, which is the same as Shatner's), and destroyed on April 29, 2285. That's about a month between movies, which seems about right to me.