For the governement tool :
I see one interesting point : there is a map showing the distribution accross the votes : http://petitionmap.unboxedconsulting.com/?petition=131215.
We can see here that the votes have been made accross all the country, and the distribution seems to follow ppopulation density (more votes on London,...). An elaborate bot could have mimic that, but that fast ? This seems unlikely.
This will still require to provide 4Billions of different mail addresses. Most of provider perform checks, and if you use one or two mail domain that aren't widely used and protected against bots, you can still detect and discards those votes.
Furthermore it's a government tools, so they can have quite some checks around it, (ban proxy IP by detecting lot of same IP, ...) but i don't know them.
So i think this petition using the government tool seems like trustable as generating so many votes considering , mail addresses, postal code repartition seems hard to do in the short amount of time it has been since the first results.
EDIT : Answering to D.W comments : since the OP is wonering if the government tool is reliable regarding the one petition with 4000000 votes, what i was saying there is in this case, it is definitively reliable, fakes votes will be filtered well enough to not be so representative.
Now if you asked for a 10k votes it's possible that fakes votes can be too representative to be trustworthy.