It's possible they could extend negotiations. As long as the EU and UK parliaments both agree to do that, they can do whatever they like, as they can just change the law to support whatever they're trying to do.
Whether there will be enough public support to do that is another matter entirely. Perhaps there will be another general election or referendum which will muddy the waters.
If the last year has taught us anything, it's that the world is a very uncertain place right now, and all predictions are worthless.
Well, that may be how the UK sees it but I can pretty much guarantee that that is not how the rest of the EU will see things. The automatic end to the negotiations and finalization after two years is currently the law, it would probably be best to go on the assumption that that's exactly how it will play out. There is some sense on the UK side that the UK will come out of the negotiations in a better position than what they have right now in the EU, but the European side is adamant that no matter what the UK will not come out of this better than when they were a part of the EU, and this makes good sense (otherwise, why would any nation remain part of the bloc).
Also, the UK has just handed the EU all their strong cards in the negotiations, and if the Scots exit the union that position will get weaker still.
Keep in mind that the exceptional situation the UK has (had?) in the EU was always an annoyance to other member states but was tolerated in order to keep the UK in the EU. If the UK falls apart and Scotland re-joins (or even remains) then the UK negotiation position with respect to those exceptions has evaporated, something that would definitely be appreciated in other EU countries.
Other general elections of referenda will not muddy the waters from an EU perspective, the deed is done, any regret will now take the form of a re-application, which will likely include the UK giving up their own currency, becoming a part of the Schengen area and getting rid of all the exceptions that were made for the UK.
This was a dumb move if there ever was one, such major decisions should require a supermajority, not a simple majority and May did a huge dis-service to the UK just now. But by the time the chickens come home to roost she'll be enjoying her pension sitting under banner reading 'I gave the people of England what they asked for, not what they needed'.
> As long as the EU and UK parliaments both agree to do that
Under Article 50(3), extensions can only be granted by unanimity in the Council. The EP and Council both need to agree to a final agreement, but the EP doesn't seem to be involved in extensions at all.
Whether there will be enough public support to do that is another matter entirely. Perhaps there will be another general election or referendum which will muddy the waters.
If the last year has taught us anything, it's that the world is a very uncertain place right now, and all predictions are worthless.