Contrary to popular belief, the UK does have a written constitution—it's just scattered across roughly 25 different pieces of legislation, subject to amendment on the fly whenever Parliament damn well pleases.
And since devolution came in, more than one parliament has to be convinced to amend the constitutional framework before it can be changed.
It is becoming apparent that The Scottish Parliament and Northern Ireland Assembly may have veto power over BRexit per the House of Lords European Union Committee (11th Report of Session 2015-16,
"The process of withdrawing from the European Union"). See paras 70-71, "The role of the devolved legislatures in implementing the withdrawal agreement" -- section 29 of the Scotland Act 1998 binds the Scottish Parliament to act in a manner compatible with EU law, and Scottish parliamentary consent would be required to amend this. (A similar provision underpins the devolution settlements of Wales—which voted for Brexit—and Northern Ireland—which voted against it.)
So we have a royal mess coming down the pipeline.
Firstly, the referendum is non-binding on parliament. Voting "leave" did not automatically trigger UK departure from the EU, it just sent the sitting parliament a strong demand signal. It's up to them to trigger Article 50 of the Lisbon Treaty, or not, in a monumentally stupid game of international diplomatic chicken. (Also, a large majority of MPs in the House of Commons are actively opposed to Brexit. Absent the referendum, a free vote on Brexit in the Commons would have been defeated by a 2:1 majority.)
Secondly, both Scotland and NI voted to stay: in the case of Scotland by a stomping 62/38 margin. The European Communities Act 1972 is effectively baked into Scottish constitutional law, per the House of Lords report, and can't be amended without the active cooperation of the Scottish parliament. Trying to override this in Westminster would trigger a new and excitingly different constitutional crisis and almost certainly lead to Scottish secession on the fast track.
Meanwhile, Scotland is already lobbying the European Commission to protect Scotland's EU membership, and it looks likely that right now the final say in whether Brexit happens lies with Nicola Sturgeon, who is First Minister of a nation that voted to stay (and leads a strongly pro-European government). Taking Scotland out of the EU against the will of the voters and their elected government would also put Scottish independence back on the fast track—and this time previously staunch supporters of the union such as J. K. Rowling are already changing their tune.
I now confess to having run out of clues. I have got no idea where this is all going to end up. If the next leader of the Conservative Party in Westminster (presumptively Bojo, although I am having nightmares about Theresa May getting the job) wants a fig-leaf for switching to "remain", Oor Nicola is about the best that they could hope for. On the other hand, if the Commission are serious about wanting the UK out, they could insist on keeping Scotland as a separate member state, just to add to the pain. The possibilities are endless, within limits. I do not expect the Queen to stick her finger in the buzzing, sparking, shorting constitutional mains socket: she's not that stupid. But that's about all I can rule out at this point.