Not OP but my partner lived in Copenhagen for two years. As with all of Scandinavia fluency in English is very common, practically universal in cities so you can happily live there without speaking a word of the local language. Also unlike most of Europe it's a lot more acceptable to just assume English is spoken and lead with that rather than stumbling through in the native language to start with.

That said it's still polite to learn some basics, e.g. thank you to mix in. If you're planning on living there long term you will want to learn Danish for social purposes unless you're planning to mix exclusively with immigrant communities. It's a very international place so English can often be the default language that any given group will all speak.

Edit: oh and an interesting quirk English is often spoken with an American accent! I think because they watch a lot of American TV shows and movies.