I think every school is different, and it also depends on which coach leaves/gets fired.
One head coach(HC) liked my son and with the pitching coach(PC) said they were going to offer scholarship money. The HC winds up getting fired, but the PC was still there and spoke to the new HC on my sons behalf. My son performs for this new HC with the same velo and stuff as he had shown before. The new HC sits down with us and says he has a new philosophy, and that to receive scholly money, RHP's need to be throwing 92 and above, and LHP need to be at 90 and above. Since my son had only topped out at 88, we were offered a preferred walk on with a guaranteed spot. The only trouble was that this school cost $50k+, so it went from being my sons first choice to off the list.
We had another school put together an athletic and academic package by the assistant HC/recruiting coordinator. It was a nice offer and we were seriously considering it. Sure enough that coach quit to take a job at another school. When we spoke to the HC, he said he had heard about my son and authorized the offer, but had never seen him pitch. So since the other coach was gone, we would need to have himself or the new recruiting coach see him first.
Needless to say, both situations were very disappointing. The first one eventually caused the PC to quit. It was not over my son specifically, but because he had a very different philosophy than the new HC about strict velo numbers equaling scholly money. Still he defended the guy because as he put it, the new HC was under the gun to produce right away or his job was in jeopardy.
Hopefully in your sons case the new coach will honor the old coaches agreement to at least give him a spot if he performs well in the tryout or Fall ball. It would really be a shame for your son to get stuck in a school he was going to play for only to lose a year and have to start looking all over again.
As others have said, get in touch with the new coach ASAP and try to establish a rapport with him. Let him know what the other coach had promised and that your son made a commitment to go there on that promise. Hopefully that will have some sway since it is a walk on, vs. having to give money based on a former coaches evaluation. You would think that without money being involved, it would be the honorable thing to do.