I see it as a step-up process. Gradual for some and quick for others.
You start with the "deciding my name" step. You'll think about what ifs and finding what works. For some, this can be a short step for a variety of reasons. Maybe you already had a name picked out, maybe you always went by a nickname and just changed to that, or whatnot. It's also a step you may go back to many times before finalization.
This step isn't essential but many of those still coming to terms with their status and/or stealth often start using the name in their head or privately as practice. They may start using it as a name when online in forums, they may make up conversations in their head using the name, or there might even be embarrassing moments where they say the name aloud or at a mirror. You can call this the "getting to know thyself" step.
The next step is actually starting to use it publicly, usually only with friends/family but it's again not required. They may ask them to call them by this name or use a nickname of this name. It's often at this time that the person asks to use different pronouns than they are used to. There may also be regression if the reception of this is very bad but many continue forward. This step I've called the "testing the waters" step.
Finally, the person is almost entirely known by their new name and/or their pronouns. This might be the point legal considerations are being decided or made. There's still a possibility the person may toss out their new name and start over but it's probably not common enough to be notable. Wrapping up, this is the "congrats to your new name" step.
Again, this isn't the only way but I notice this trend of trans* people changing their names. It isn't also quick but it isn't always a slow process.
