|To say Wolf 359 was a disaster is like saying the Battle of Tsushima was a misstep for the Russian Empire.|
(via Memory Alpha)
I can imagine that there would be a inquiry into how exactly the massacre happened. You'd figure that 40 ships would be more than enough to take down even an all-powerful Borg cube. Memory Alpha's article on the battle says that the Federation fleet attacked in small groups instead of one all-out mass assault and that allowed the Borg to destroy the fleet in short order. I see the inquiry having one of two outcomes:
1. Hanson, the admiral in charge of the fleet at Wolf 359 is blamed for the disaster and he goes down in history as one of the worst, if not the worst, naval officers in history.
|A man out of his depth who left a tarnished legacy.|
(via Memory Alpha)
Of the two, I see the first as being the most likely. Picard would undoubtedly be blamed by some both in Starfleet and in the public/political sphere, but I think he would be resolved of any real culpability since he was assimilated at the time and forced to help the Borg against his will. Going further on that, I think there would probably be those high up in Starfleet who would be uncomfortable with Picard in the aftermath and would want to either push him into retirement or transfer him away from starship command into a desk job, possibly in some corner of the Federation. Certainly I think that his reputation with both Starfleet and the public would be hugely tarnished. We saw evidence of this in First Contact when the Enterprise-E is sent to patrol the Neutral Zone instead of helping to fight the Borg incursion. There was a lack of trust from the admiralty in Picard when it came to the Borg, almost as if they were afraid that he'd relapse into Locutus.
On the other hand, I think Picard managed to rebuild his reputation within Starfleet by the time of First Contact seeing as how the fleet readily accepted his taking command upon the Enterprise's arrival.
I think the other big thing to happen post-Wolf 359 is that there would be a huge shakeup in Starfleet itself. To put it mildly, Starfleet got caught with its polyester pants down. Throughout TNG and Deep Space Nine we saw that the bulk of the Federation fleet appeared to be Miranda and Excelsior classes from the TOS movies. We saw very few newer ships during TNG's run and it wasn't until First Contact that we saw a whole batch of new ones. Even then, we still saw mostly the old TOS era ships during most of the battles in the Dominion War. I think the development of the Sovereign, Akira, Steamrunner, and other ship classes were a direct consequence of Wolf 359. We know the Defiant-class was designed specifically to fight the Borg, so it's not a stretch that the battle kickstarted a shipbuilding and modernization program.
|The Defiant and Akira classes were presumed|
fruits of a post-Wolf 359 modernization program.
(via Memory Alpha)
And that complacency lead to the deaths of nearly eleven thousand people and the destruction of 39 ships. So yeah, heads probably rolled at Starfleet HQ. I think that there would be a shift in the mindset of Starfleet, with the defense hawks taking on more prominent positions at HQ. These incoming hawks would, I imagine, be veterans of the Cardassian War and those other mentioned but never elaborated conflicts. Starfleet doesn't become a military organization, but there's definitely a shift towards defense. The scale of destruction drives home how under defended the Federation was and that relying on nearly century old starships isn't the best of ideas. So still a scientific and exploration dedicated organization but with an increased emphasis on defense.
What's truly amazing about the Battle of Wolf 359 is that the Borg actually saved the Federation in the long run. The assumed changes that occurred in the aftermath helped prepare Starfleet for the Dominion War that came along years later on Deep Space Nine. It's likely that if the curbstomping hadn't happened, the Federation wouldn't have survived the later conflict.