If Theresa May resigns
The Conservative party will hold a leadership election to replace Theresa May. This process would takes 6 to 8 weeks, or, as last time when PM May was chosen, the new prime minister could be appointed by acclamation if all the other candidates all backed out, which could take a week or 10 days.
If she does not resign
Presuming she survives Corbyn's no-confidence motion, the Prime Minister could soften her deal to win over the significant number of Labour MPs who favour something more along the lines of Norway plus. However, this scenario raises various questions:
Would Jeremy Corbyn, who has thus far avoided pinning his colours to any particular policy, be willing to get behind a softer Brexit and risk angering both Labour Leave voters and the party's grassroots, who are calling for a second referendum?
Would PM May be able to secure a special permanent customs union arrangement tailored to the needs of Great Britain, as proposed by Mr. Corbyn; EU officials have signalled the EU would be willing to talk on this.
Brexiteers fear that for regulatory alignment reasons, the price for staying in the customs union will also be staying signed up to freedom of movement, so Britain would not gain control of its immigration policy.
The DUP might walk out of the Government over the Northern Ireland backstop, causing the latter's collapse .
Brexit as a No-Deal exit
There is currently no majority for no-deal in Parliament, however the Prime Minister could opt for the 'nuclear option' - attaching a vote of confidence in the Government to the no-deal vote. In other words, voting down no-deal would automatically lead to the collapse of the Government. After all, how many are genuinely willing to bring down their own administration, risk losing their seats in a chaotic general election, or expose themselves to possible deselection for stopping Brexit, in order to prevent a no-deal?
With support from Tory Brexiteers, more ambivalent Tories, Leave-backing Labour MPs like Kate Hoey, as well as the 10 DUP MPs (who have in the past asserted that “no deal is better than a bad deal” and a no deal is “probably inevitable"), no-deal could scrape together just enough support to get across the line.
A typical strategy of the Prime Minsiter is to run down the clock. Her Plan B may thus be to allow Parliament to debate and vote on all the options on the table in turn, in a series of so-called ‘indicative’ votes - namely EEA membership, no deal, a Second Referendum, and her own deal (perhaps whilst seeking to renegotiate aspects of it with the EU).
Labour’s No Confidence Motion
Jeremy Corbyn tabled a no-confidence motion in the Government within minutes of the parliamentary vote results. The vote could cause PM May to resign, which could cause the Tories to go into further meltdown. The downside for Mr. Corbyn is that Labour seems unlikely to win a no-confidence motion, as the DUP has given no indication that it would ever vote in a way that would help a Corbyn government to power. Once a confidence vote is out of the way, the Labour grassroots would call for Mr. Corbyn to get behind a second referendum, anticipating a Remain result.
If Labour does win a no-confidence motion, then the race for the Conservatives to try and form another Government that commands the confidence of the House begins. If a motion of confidence stating "That this House has confidence in Her Majesty's Government", is not passed within 14 days then a general election must be called.
No Brexit by revoking Article 50
Remainers who would like to stop Brexit without having to bother with the messy, democratic business of a Second Referendum prefer this option. Moderate MPs might vote to revoke Article 50 as a means of stopping the clock to allow renegotiation of the deal, or more time to prepare for a No Deal. The problem is that the EU may be able to veto the revocation if they suspect "abusive practice" – and using the tool to stop the clock, or buy the government more time might be deemed as such. If the latter is the case, then it's hard to see how hardline Remainers could win over more moderate MPs nervous about being seen as stealing Brexit from the people.