Thanks! Corrected. You're right that it isn't equivalent. There is no substantial effort afaik to only rid one of the branches of government (executive, legislative, judicial), so the secession movement is the closest I can think of to the UK's Republican party, which really isn't that close at all.
The UK's republican movement doesn’t want to get rid off a branch of government, but merely wants to change how the person to occupy a certain position is chosen. The actual effect of abolishing the UK monarchy on politics would be close to zero.
The Queen still has extensive powers, that she does not use.
For example, all laws must be signed by the Queen, but the last time someone refused to sign a law was around the 1700s, but if she wants, she can still refuse to sign a law.
Also, the Queen can refuse a parliament election (resulting into having no parliament)
Also, she can sell whatever she wants that belongs to the state in a unilateral manner (but I do not know the last time a monarch used this power O.o)
Also she can kick out the prime minister (this power seemly was threatened to be used on Tony Blair when he got excessively trigger happy regarding US wars... seemly threats to use it, made him tone down a bit his hunger for blood).