I'm just going to chime in and say that I've been dealing with the same issue. The problem appears to be experienced by fast typers. Maybe include an experimental feature to increase the type buffer for fast typers, just a simple toggle, or maybe make it a time interval based on how long it takes generally for Cortana to pop up instead of how long you think it takes the average person to type a letter. Should be less variability in Cortana response time than user type-speed. Just giving some ability to adjust would make a huge difference. Maybe find a way to poll Cortana once, capture the response time in ms and adjust your buffer time accordingly. This is all possible in C++ or Java or whatever. It's not good to have hardcoded workarounds where they need to be flexible. If this sounds too complicated for the average user to want to mess with then add it to an "advanced search functions" window.
I would prefer to use Start10's search function, but it misses many files and programs on the system. If I have a program that is portable then it's missed for instance. With Cortana search I can pretty much reliably search for anything I'm thinking of on my system (although curiously sometimes it misses newly installed programs I haven't run yet).
Jankiest solution, but better than nothing: Add "Something missing? Try Cortana" to Start10's search. So if I search with Start10's search and don't get the results I'm looking for I can just tap that button and it takes my entire query and pops it into Cortana. I'd rather have 1 extra mouse click than 5-10 backspaces every time I type a search, or have to remember another keyboard shortcut.