In British English, which is more common — the em dash or the en dash?

This is a matter of style, so it's not possible to give a definitive answer on what the correct use is. Different style guides, and different people, will use dashes in different ways.

Having said that, it's generally been the case that more British style guides will say to not use an em dash but, where US style would use an em dash, to use an en dash that's surrounded by spaces.

From the University of Oxford Style Guide (PDF), page 13:

m-dash (—)
Do not use; use an n-dash instead.

n-dash (–)
Use in a pair in place of round brackets or commas, surrounded by spaces.
      ✔ It was – as far as I could tell – the only example of its kind.
      ✔ The library – which was built in the seventeenth century – needs to be repaired.

Use singly and surrounded by spaces to link two parts of a sentence, in place of a colon.
      ✔ The bus was late today – we nearly missed the lecture.

Use to link concepts or ranges of numbers, with no spaces either side.
      ✔ German–Polish non-aggression pact
      ✔ The salary for the post is £25,000–£30,000.
      ✔ Radio 1 is aimed at the 18–25 age bracket.

Use between names of joint authors/creators/performers etc to distinguish from hyphenated names of a single person.
      ✔ Lennon–McCartney compositions
      ✔ Superman–Batman crossover comics

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Note that the guidance here to not use the em dash goes against the guidance of most US-based style guides. (But also note that the use of the en dash recommended in the last two categories—without a surrounding space—does match the use of the en dash recommended by most US style guides.)

But that is only one of the common style guides used in the UK—and many companies and people in the UK do use em dashes. So, it should not be thought of as definitive. As with other aspects of style, pick the style guide that is being used by your audience. If there isn't one, then pick the one you like—or make up your own style sheet from a combination of style guides. Just be consistent.