Constitutionally, moving to a Republic model would be a nightmare. The power of Parliament and legislation and all government institutions lies, theoretically within the corpus of the Monarch as a conduit from God.
You would have to unravel it all, and reorientate the locus of power, and deal with all legislation prior to that point, which would become void. It was too much to face back in the 17th century, which is why Cromwell was pretty much a replacement King and why, after his death, we ended up with the restoration.
What's interesting is that most Monarchies end through revolution, at a time when the fabric of the state is irretrievably broken. That's because it's one of the only sitiations by which such a shift can occur without a horrific mess - - at a point where you have a year zero, more or less.
Put it this way, constitutional experts stated that Scottish Independence would likely take ten to fifteen years to sort out because it would void the Act of Union, and a hellova lot of subsequent legislation depends on that Act.
By abolishing the Monarchy, you are essentially looking at removing the power foundation for every piece of British/English legislation.
Put it this way, if the entire British Royal Family was erradicated by aliens, and we had no Monarch, Parliament couldn't pass law, the Police and Armed Forces would have no jurisdiction etc. That's why the transfer of power is instantaneous on death to the next Monarch. If there is no new Monarch, everything stops.
And you have to be careful what you wish for. A translation mistake in the Great Reform Act led to female heads of household losing the right to vote, and that wasn't fixed for nearly seventy years.
And I'm not particularly a Royalist either, yet I'd still urge caution. Would the armed forces accept a transfer of sovereignty to an elected president? You would be changing the entire dynamic of those institutions by doing so. Ditto the civil service and the Police.
The Monarchy provides stability; in fact it provides so much stability, most of us take it for granted and think it's normal.
And to be honest, with how complicated the British state has become in the last 100 years, I would really question anyone who wanted to be president. Being Prime Minister is enough of a shitshow as it is.