Keep - for now - of for no other reason than if Brexit was a shitstorm of piss-poor deal and rewritten laws, removing the Monarchy would make it seem like a storm in a teacup.
The presence of the Royal Family is woven into every part of our constitution and laws. We would be completely starting from scratch on everything. You can't abolish the Queen overnight and 'just keep' Parliament with a different title, because Parliament only has power in conjunction with the involvement of the House of Lords and the Courts.
But, if you remove and abolish the Monarchy, then you remove and abolish the House of Lords (and potentially immediately, depending on how the Letters Patent are ultimately viewed; the Queen is the start-point for all titles and the legitimacy of them.) You definitely have no 'new' Lords, either created or Hereditary - noble titles don't just 'pass down' automatically; the Patent has to be reconfirmed each time by the current Monarch - which sounds like its not an issue until it becomes a lot of power concentrated in an ever-smaller and aging body.
You need a whole new Constitution to get away from that issue.
The Courts also suddenly have major issues. If we unpick such huge sections of our legal framework, then massive, massive amounts of the rest of UK law and legal precedent is called into question. Is a law signed into power (and any judgements made under it) valid when the people who signed it in have been made abeyant? It's certainly a cause for a lack of clarity.
The armed forces, the Treasury, the Courts - all would need overhauls to their regulations and procedures.
Bear in mind, too - it isn't just us. The Queen is head of the Commonwealth - and the head of the Church of England.
Then, you'd have the fights over who owns what/whose liable for what/public/private/in trust etc - they'd be horrendous.
That's without the optics and the intangibles. The Queen is a figure of continuity and stability. Much of that goes with her death, definitely, but the continuing nature of the monarchy has an influence. Watch what happens to the FTSE around reports of her being unwell, or even Prince Phillip. Her death will impact the value of the pound. The country simply, at the current time, and probably for at least the next generation, can't take a shit storm of uncertainty of that magnitude. Even then, its the work of decades to transition away.
This is not to say it shouldn't be something to look at in the future, or that time for it (certainly in its current form) isn't over, but its not so simple as people would like it to be, and it isn't as easy as saying 'get rid of them all'.
It may well become academic. Charles is very clearly in favour of a much trimmed down take on it all. He's a large proponent of the 'Royal Family' being the immediate line of succession only and really then only the first few on the list. With the Queen's death, that will become him, William, and William's three children. He definitely intends to cut out the more extended family in the form of Andrew, Edward and the rest.
Even Harry leaving, I highly doubt he resisted it - he may even have encouraged the idea - and there's already a fair bit of speculation that Harry hasn't already formally abdicated his place in the line of succession only because William's kids are so young and Charles is already a relatively old man. Should something happen to William, then there's a very real risk of a time where George is still a minor on his grandfather's death. In that event, he needs a Regent, which Harry can't be if he abdicates.
I also suspect that Charles will 'retire' at some point, as the modern Danish Royals do, and that by the time we're moving to the William to George transition, the monarchy will be a very much different institution in any case. It may be that they know which way its going, and that they'll start to move to a more modern 'set up' on the death of the Queen without any need to 'abolish'.