I had the same issue.
I couldn't understand it with dependency walker, as even working apps showed missing dlls.
I used qwt in my program and compiled it in Qt 5.5 with MSVC2013 32bit on a 64bit Windows 10.
After changing 'Qt' folder name to 'QtHidden', as sugested here, I attempted to get the application to run with all the dlls in the same folder and was getting an weird error about qwt like you, althouth I had added qwt.dll.
To get it working I did the following:
Copied all dlls in QtHidden/5.5/mingw492_32/bin to my deploying folder
I copied all folders in QtHidden/5.5/mingw492_32/plugins
Copied all folders in QtHidden/5.5/mingw492_32/qml
// Pretty standard untill now
added qwt.dll
// Now for the trick
Performed the steps 1,2,3 but for the folder QtHidden/5.5/msvc2013, overwriting all dlls and folders previously added from /mingw492_32.
So, what I believe is happening is that some dlls from mingw are needed, like libgcc_s_dw2-1.dll and libstdc++-6.dll, but all other dlls need to come from the MSVC2013 folder for qwt to work, wich was the compiler used.
I managed to run it in Windows Xp 32bit after adding MSVCP120.dll and MSVCR120.dll. Still needed this extra ones because MSVC2013 wasn't installed in the Windows Xp computer.
Compiled in Windows 10 with all resources and ran on Windows Xp. I'd call it a success.
Hope it helps someone else.