During World War II, the British used the phrase "no man is an island" to justify the fight against Nazi Germany. Would John Donne approve of this?

Yes, John Donne would approve this. The phrase comes from one of his sermons, Meditation XVII. The quote is as follows:

No man is an island entire of itself; every man is a piece of the continent, a part of the main...

What Donne means by this is we are all in this life together. No person is a self-contained unit, apart from everyone else. Instead, we are all mutually interdependent. We are all part of a larger community.
Donne goes on to say if one of us dies, the rest of us are diminished. We all need each other.
The British used this idea to support militarily helping their neighboring countries overrun by Hitler and the Nazis during World War II. Britain, in fact, was and is an island, but the British understood they could not act as if they were apart from the rest of Europe. What happened in other countries affected them greatly. Their willingness to join in the fight is the opposite of the foreign policy doctrine of isolationism, which is to decide that the problems of other nations are not ours and to turn away from helping them.