There is absolutely no indication that there was a passage of days anywhere in the movie. The closest we get is this:
Star Trek: First Contact wrote:
PICARD: One, and it's on a direct course for Earth. It'll cross the Federation border in less than an hour. Admiral Hayes is mobilizing a fleet in the Typhon sector.
DATA: At maximum warp it will take us three hours twenty-five minutes.
Just because that is where they are meeting up doesn't mean that is where the battle with the Borg takes place. There's no indication that a group of starships could hold off the Borg for days. Plus, it makes no sense that the Enterprise would be at red alert for that long.
Just like the Abrams films, they compressed the time frame to keep everything up-tempo.
Hang on, hang on...
The line immdeiately after that is Picard inturrupting and saying, "We aren't going." So, the Enterprise is three and a half hours away from a battle in the Typhon sector, but they don't go. Later, we see the Enterprise join the battle at Earth. So the battle must have moved from the Typhon sector to Earth. I doubt the Borg would have been nice and friendly and not shot anyone while they were all going to Earth, so that indicates to me there was a running battle between the Typhon sector and earth.
Now, how long did it take? Well, considering that the Bozeman was lost in the Typhon expanse for a century or so before anyone went looking, it's safe to say that the Typhon sector is not near Earth. Out in the sticks is far more likely. So it's going to be some distance from Earth. Probably take a few days to cover the distance.
Thus, a running battle for a few days as the Borg went through the Typhon sector, continued to Earth, and the fleet tried to get them to stop by constantly attacking them.
Now, it is possible that the fleet met in the Typhon sector and then proceded to Earth, but why? Why assemble your fleet in an out of the way place like that?