It's commonly used in spoken English. Sometimes this phrase is shortened to "long story short".

We could discuss its origins and so forth. But... "to cut a long story short" it is commonly used in English.

As an American, I'm used to hearing "to make a long story short". We say this when we don't want to go into all the details but just want to summarize the results. What happened on 9/11/2001? To make a long story short, some crazy people flew some airplanes into some buildings and killed a lot of innocent people for no reason.

In U.S. English, written and spoken, the phrase "to make a long story short" is very common. As TeacherJan notes, in the last decade or so it's become common to say simply "long story short." "To cut a long story short" is also common, but less so. Some actual examples: "Long story short, Don Quixote' is the tale of an eccentric man who, after reading a book about ancient chivalry, embarks on a quest to find his ideal woman."--Huffington Post, 2016 "To make a long story short, in an online naming contest for a new ice-strengthened polar research vessel being built for the British Antarctic Survey, ['Boaty McBoatface'] was winning out over more, er, serious ones — like 'RSS Henry Worsley' or 'RSS David Attenborough.'"--The Washington Post, 2016 "Anderson Cooper asked Trump to defend his recent spat with opponent Ted Cruz—which, to cut a long story short, started as a fight over some pictures of Melania Trump in British GQ, and quickly devolved into a competition over whose wife is hotter."--GQ magazine, 2016

As mentioned, it is common (from my experience, it seems "to make a long story short" is more popular). A dictionary gives the example of "I got fired". Being fired is big, bad news, and usually has a story behind it which might be long and involved. So if you want to break this news to a friend skipping the detail of what happened, you might say: Long story short, I got fired. It would make it smoother than just saying "I got fired".