Per the FDA website, they will respond to breakthrough applications within 60 days of receiving them: 

What are the timelines for FDA to respond to a breakthrough therapy designation request? 

FDA will respond to breakthrough therapy designation requests within 60 days of receipt of the request. 

https://www.fda.gov/regulatory-information/food-and-drug-administration-safety-and-innovation-act-fdasia/frequently-asked-questions-breakthrough-therapies

DM's timeline from the CTAD CC:

"As we’ve mentioned before, we’re applying for breakthrough therapy status with both the FDA and the EMA. The process will take 90 to 120 days". 

I can't tell, either, whether that timeline includes the FDA's 60 days to decide or is in addition to it.