https://en.wikipedia.org/wiki/Category:Rival_successions
Like there can be some guy living in another country who says "um, that monarchy no longer even exists" or "um, my ancestors lost that war a long time ago and I don't plan to reignite it", but still has some kind of association of enthusiasts who claim to actively want him on the throne. One such example is https://en.wikipedia.org/wiki/Franz,_Duke_of_Bavaria who insists that he doesn't actually think of himself as the King of England -- his relative who last actually exercised that position was deposed back in 1688 -- but that doesn't stop other people from thinking of him that way. :-)
Wikipedia has several dozen people listed in the official present-day line of succession to the British throne and notes that a genealogist has claimed to identify over 1,000 people in that line.
It's possible to imagine a France-like republican future for Britain in which people's elaborately detailed and extensively researched pretense claims are not very interesting news for the overwhelming majority of the population, but it probably wouldn't be something that would happen overnight in response to a couple of abdications when a huge number of British people and a substantial number of people with claims to the throne are actively enthusiastic about the monarchy.
Monarchy isn't a hot potato, that the current power-holder has to directly pass on lest it get "dropped on the floor" and cease to exist. Monarch is a belief of the people that a position of power exists and is "open" to be taken by a person with a claim on "right to rule." It's a thing you can convince people you deserve, at which point they put the crown on your head rather than shooting at you for wearing it. Succession gives a peaceful path for that transfer of power, but succession isn't the default; a power vacuum being grabbed at by random nobles—usually violently!—is.
If the current monarch stepped down, someone else without the same scruples would just step up. You'd have to, in effect, get every even-barely-theoretically-"righteous" noble to all agree to denounce the power, for the power to truly go away.