When the development studio stops replying
The development studio stops replying. Not dramatically. The updates get shorter, the calls get rescheduled, and then there is nothing.
What decides how bad the next month is has almost nothing to do with the code. It is whether you can hand what exists to someone else.
Four things determine that, and all four are free to establish while everyone is still friendly. The source code is in a repository your company owns, not one belonging to the studio. The accounts that host and run the thing are in your name, with your billing, and you can log into them today. Someone other than the original developer has started it up successfully at least once. And there is a written description of how it deploys that is not just in one person's head.
None of that is a trust issue and you should not frame it as one. Studios close, get acquired, and lose the one engineer who knew your project.
Ask for all four in the first month, not the month you need them.