Re: jaws & NVDA in combination
Has someone customized your JAWS installation as far as keystrokes go in some way? The typical keystroke set to toggle speech on/off in JAWS (at least in the versions I've been tutoring, which goes back to JAWS 14) is INS+SPACEBAR, followed by S (substitute CAPS LOCK for INS if you're using laptop keyboard layout) and I've never had INS+S toggle speech in JAWS (but that's the toggle [or switcher, in default mode] in NVDA).
I do, however, use the NoBeepsSpeechMode add-on in NVDA so that INS+S is a true toggle on/off for speech rather than a three-way switcher.
Another option would be trying the "Pass Key Through" command in either JAWS [INS+3 on the number row] or NVDA [NVDA+F2], depending on which needs to be ignoring the next keystroke and passing it along to the other, before you hit the INS+S toggle. I imagine order of processing precedence will be determined by which screen reader has focus at that moment, JAWS versus NVDA, but you'd have to play with this technique to test that out.