If it's on your side of the boundary line then no but in the spirit of good neighbourliness it may be prudent to explain your planned reduction to your neighbour BUT if it a shared boundary then you must discuss the matter with your neighbour as you are both responsible for it.

Whose hedge is it? If its yours you can do what you like with it. If its your neighbour's you shouldn't do anything with it except you can trim the parts that overhang the boundary.
If neither of you knows whose it is you need to come to an agreement.
As Barbara says in any case it would be polite to mention it, though at 2m it shouldn't affect their privacy when it has recovered enough to grow shoots lower down..

You ask a question which involves an aspect of law (though not a formal one ) called “neighbor law”. If you cannot approach your neighbor about this significant reduction in height I would suggest you contact a lawyer familiar with the subject of neighborly rights and restrictions. Example.....doing something on your property that might affect your neighbor’s enjoyment of their property such as obstruction of a view, or interference with their privacy or the big complaint these days, cigarette smoke blowing onto their property.

That information is often on your deeds if you have access to them. Uk law is a bit different from Loosestrife's in the US and first complaints are usually made to the local council. We don't have a law called Neighbour Law in the UK, though it sounds lie a good idea..and you are entitled to cut back or remove a hedge of your own without any permission.

Since March 2013 the law in Scotland changed. see the link. https://www.telegraph.co.uk/news/uknews/scotland/9960521/High-hedges-laws-passed-at-Scottish-Parliament.html
The law in Wales was changed in 2004 see the link
https://www.planningportal.co.uk/wales_en/info/3/common_projects/25/high_hedges
The law in England is covered by the legislation in the following discussion. http://naturenet.net/trees/hedgerow/highhedges.html
As you will see the law is generally the same throughout the UK with some differences. All recommend a decision made between the neighbours is the best and least expensive option. No guarantees are given on who might get a successful outcome if you allow the relevant authority to rule on the matter.

Totally agree - I was thinking about what to do if the neighbours can't come to an agreement. But that would only come into play if one neighbour wanted to complain about the others' hedge. If its your own hedge you can cut it down anyway.

By common law, no you technically don't need permission.
By the laws of common courtesy, yes you do. Which law you follow is on you, but consider the predictable out come. Then there's the 'Golden Rule of Life,' do unto others as you have them do to you - you reap what you sow.