2nd unit will continue to film scenes and interstitials the film needs. 1st unit is the primary unit that Bay directs from and works directly with the actors. Generally if you see a movie's lead on screen, chances are its 1st unit work.
I guess Bay's statement that the injury will not cause delays was before the schedule and what was left to shoot was given the once over. Considering Shia's injuries pretty much every scene he is probably had to be re-designed to cover extra use of stuntmen and to either hide his hand or work around the injury.
An example of the level of difficulty is the Barricade scene. Technically not much stunt work was needed but during that scene Shia used his hands constantly to push himself off cars, crawl back, slam against the police cruiser and so forth. With his current injury he simple would not have been able to film that. Instead the scene might be covered in a different way, more distance or close-up with a stuntman doing the bumping into the cars, crawling up the hood and charging from the lot. Problem with that though is it means more behind the head shots, less reaction shots and the like.
Now that is just once scene but carry that out through the entire movie (say something as simple as holding the All Spark) and you get the idea of how a hand injury can complicate things. There is a reason why even when films do the fake injuries (shot shoulder, hurt leg, etc) they tend to leave the hands alone. Considering that it is safe to assume the 2nd film will up the ante across the board, that also ups the complexity of trying to shoot around that hand regardless if the injury is added to the story or not.
Will this delay the movie release next June? I doubt it. Bay started work on the sequel pretty much immediately, working hand in hand with ILM and Digital Domain, giving the entire production a substantial head start. I think there is still about a 3-4 month buffer in the schedule before the threat of film release delay becomes a danger. In the meantime, best wishes to Shia and good luck in the recovery.
Source: Transformers Live