As crmprogdev mentioned, you'll want to go the Visual Flow route (Triggered/Called by Process Builder). I'll provide a bit more detail on the how-to (based just on what you've said).
You're referring to an issue that happens when the fields typically aren't yet set and the Process Builder is trying to fire. The best way is for us to fire the Flow from the Process Builder and then do everything in our Flow.
You're going to be able to accomplish this with most likely just two elements in your Flow. One to lookup the Transcript record and the other to update your Case.
First things first: Create two variables
1 - var_TranscriptId [text - set to Input/Output]
2 - var_CaseId [text]
First Element: Record Lookup
Set the Object to be your Transcript. Criteria is Id = var_TranscriptId (passed in from Process Builder). In your setting values, set the Case lookup to the var_CaseId variable.
Second Element: Record Update
Set the Object to be Case. Criteria is Id = var_CaseId. In your fields to update, grab the Time-Out checkbox and then type in 'false' into the right box and you'll see the flow variable for an unchecked boolean that you can use... and just select that.
Connect the two elements, set the first as your starting element. Go to your Process Builder, keep the same criteria you had, and now instead of an update fire your Flow and pass in the TranscriptId value into the Flow.
You can also check out my site www.salesforcesidekick.com for some how-to and examples on Flow.