There is no natural way in MathML to duplicate the way that \not works in TeX, so this is one of the few places where MathJax intentionally diverges from what TeX does. In MathML, the not symbol is a character that follows the negated character and overlaps it from the right (like \lap{/} would). Conversely, TeX's \not overlaps from the left. Early versions of MathJax did overlap from the left (as you indicate), but that was changed somewhere along the way to make it work better with MathML (particularly the NativeMML output). So now \not\Longrightarrow has the slash near the right. In the case where the negated quantity is not a single character, MathJax will overlap from the left.
Note that \implies includes spacing around it, so is not a single character (it is space plus a character plus space), so MathJax overlaps from the left. But because of the extra space, it doesn't work very well. (This is also the case in LaTeX, so MathJax's output mirrors TeX in this case.) You are right that \centernot is not available.
Here is an alternative that you could try:
\mathrel{\rlap{\hskip .5em/}}\Longrightarrow
Hope that works better for you.
Davide