Like Tim pointed out, this will copy all the text files in different directories to the same destination directory. However, you were particular that you wanted to use the cp command. Also, I assume that your "txt" files are conveniently named with a .txt "extension". And, yes I cheated by using zsh.
Like Andrew suggests, I suppose one can use the file command to distinguish text from non-text files, but relying on magic isn't foolproof.
"Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away." -- Antoine de Saint-Exupery
Now here's a classic reason for not using shortcut English!
I read this as "txt" files, as in "filename.txt". I saw the "txt" and didn't notice the exact form of the wildcard, which is to scan ALL files. The problem, as has been pointed out, is MUCH bigger when you want all the text files!
For one thing, is an XML file a "text" file for the case in question?
An IDE is no substitute for an Intelligent Developer.