Unfortunately, there isn't much that you can do to speed up the process, which involves Missouri's governor authorizing the paperwork for the extradition and then dispatching a team of Marshals to come and pick him up. If Missouri isn't interested in extraditing after all, then as soon as word gets back to Alabama, he would have to be released so long as there are not also any Alabama cases holding him in. Otherwise, extradition will pretty much occur when it occurs, and there's not much which can be done unless/until the delay is unreasonable.
Basically, Missouri would have 30 days from when your fiance signed the waiver of extradition and agreed to return to Missouri, for Missouri to get their papers together and dispatch a team to come get him. That's because there's a Federal Act called the Uniform Extradition Act that says that 30 days is a reasonable amount of time for an extradition to be completed.
States are free to make their own modifications and are not bound by the Act, but because Federal Law says that 30 days is reasonable and Federal law trumps state, there isn't much one can do about a delay until after 30 days are up. Then, if there is still no movement on the part of the other state, your fiancé's lawyer can file a writ of habeas corpus to bring him before the judge for the Alabama to show good cause as to why he was still being detained and, if the state couldn't show that, to release him.
Usually judges keep close watch on extradition cases since these defendants can easily fall through the cracks of the system if there is no other case they are being held for. Your fiancé probably has a public defender assigned to him if you haven't paid for private counsel. You need to reach out and speak to him or, if he has no lawyer, you have to get a local lawyer involved. That way at the earliest reasonable point, a lawyer would be available to do a writ as I indicated before, and to argue for his release if good cause can't be shown as to why Alabama's still holding him.