The EU (Withdrawal) Act 2018 set March 29th as "Exit Day", on which day most of its provisions come into effect; most notably, the repeal of the European Communities Act 1972, which is the Act that gave legal force to the UK's joining the EEC in the first place.
This is defined in Section 20, Subsections 1 and 2 of the Act, and is primary legislation. There are therefore only two ways to change this: primary legislation, for which there is no time anyway, or statutory instrument, which would need approval from both Houses (see EU (Withdrawal) Act 2018, Section (20), Subsection (4), and Schedule 7, Section (14)).
But even then it's clearly academic: the UK is currently only able to avoid exiting from the EU on March 29th if it also either agrees to an extension with the unanimous approval of the EU27 or revokes Article 50 notification altogether. The first is becoming actually tricky to envisage:
-- what is the actual point of a short extension when there are still no signs that Parliament can agree on a given version of Brexit that avoids a No Deal exit?
-- Would a longer extension, which, among other things, implies that the UK has to elect new MEPs, be any more acceptable to Brexiters, or to the EU, who would presumably rather see us go now if we are actually going?
And the second option, revoking A50 notification altogether, would have to be a "good faith" withdrawal anyway, because if it is just a cheap way of buying two more years then that undermines the spirit of the recent ruling from the ECJ on the matter.
Anyway, the net effect is that the UK has roughly speaking a week to decide what it wants, two days to ensure that the EU is at least in principle on board with that, and then needs another week to make sure that both Houses agree to the given extension too, so that the necessary changes to the EU (Withdrawal) Act can be implemented.