And the Isle of Man and the Channel Islands are in personal union with the crown, so they would become independent.
Normally a long-drawn out monarchy doesn't leave, unless it is evicted in a revolution -- as in the French monarchy (800 years) or the Russian (400).
the british monarchy nearly bought it in the 1600s, but it survived.
There's no reason a Republican government would do better in England.
There is the also the issue of what essentially is a binding treaty with the other Commonwealth Realms that are fully independent and have their own monarchy as an institution, but using the same Royal Family to reign (example - Queen Elizabeth II is fully and independently the Queen of Canada or Queen of Jamaica) - all of them would have to agree to the changes I would think.