I think we're 5-8 years away from any large assault on the alligator population, but I don't think it will be any longer than that.
1) Florida's legislature has been pretty much perma-reded now. This is not an environmentally friendly group.
2) As the alligator population in suburban and urban Florida increases (or explodes, which it will because there's nothing to limit it) there will be more nuisance calls about them. There may not be many more attacks on people, but there will be more attacks on dogs, on cats, more alligators ending up in places people don't want them to be.
3) This will call for a rewriting of Florida laws when it comes to alligators, and allowing for more aggressive approaches to controlling them. For example, the 4-foot rule is antiquated and won't last much longer, because there's nothing in a suburban Florida lake that can kill a four foot alligator except a larger alligator. All that leaving four footers (or three, or even two) footers alone does is ensure they grow up to be larger alligators.
4) The laws won't face much Floridian opposition, because the people who live there DON'T WANT THEM IN THEIR BACKYARDS, OR EATING THEIR DOGS, OR ON THEIR GOLF COURSES OR IN THEIR SWIMMING POOLS. The laws will face federal opposition, but it's going to be hard to put them back on the endangered species list, and without that protection, the federal government is rather limited in its ability to prevent state action in things like this.
5) Once the laws are changed, cities will become majorly aggressive in alligator control. They'll kill a lot of the adults; they will search and wipe out nests. it won't just be Disney, it will be Orlando, Winter park, Lakeland, Eustis -- any and every city in Florida will aggressively attack the perceived problem. most will leave some degree of refuge, but for the most park cities will want to eradicate the nuisance whenever and where ever they can. So will individual homeowner associations and developers. Because the people who live there (see yelling above).
I don't think anything will happen in the Glades, or in rural areas of Florida. I think you might still see some migration, but the migrating alligators will be dealt with quickly and aggressively. The animals will be seen not as scenery. but as a threat. But it won't be Disney that addresses the issue, it will be the state. And I believe that is coming.