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If prospective United Kingdom MPs must swear allegiance to the monarch, and if MPs are needed to change laws, how might the UK become a republic, assuming sufficient public support?

Another revolution, obviously. They did that in 1688.

Much more likely, they would politely inform the Queen that the people don't seem to want a Queen any more. If that is really the case, and not just a vocal minority, then after some polite back and forth the Queen would probably step down rather than fighting a civil war which she is unlikely to win.

While many Brits will be true to their oath, consider the difference between the individual monarch, the Crown, and the Crown-in-Parliament (which is a polite way of pretending that the monarch is still involved in legislation).

There is the quip what when it comes to dusty, centuries-old rights and prerogatives, the British monarch gets at most one try at overriding the elected government.

There are lots of republican MPs (probably including the Leader of Her Majesty's Loyal Opposition). Some of them have taken the Loyal Oath with their fingers crossed behind their backs.

At the moment, it's not worth campaigning for a republic because polls show that the public is overwhelmingly in favour of the monarchy. That could easily change when the Queen dies. At that point (if enough MPs were republican) they'd just invite the King to step down. If he declined, we would have a constitutional crisis (my guess is that he'd be told his "advisors" had advised him to step down, so he had).

In recent years the constitutional settlement seems to be that significant changes to the constitution require a referendum.

Now even supposing that all MPs took the oath of allegiance entirely literally. There would be no breaking of this oath in legislation to enable a referendum on the abolition of the monarchy. The citizens swear no oath of loyalty and could vote in a referendum to create a monarchy without breaking any oath.

This all assumes a fiction: that the oath of allegiance would prevent MPs from doing anything. In reality, the Crown must do as it is instructed by Parliament. If Parliament says "There is no Monarch" then that is the law, there is no question that a republican Parliament would not act based only on the oath of allegiance. Don't forget that many of the founding fathers of the USA had, at one point or another, sworn allegiance to the King. If the politics require it, the oath is pretty meaningless.