We use context to separate Noel (name) from Noel (crimble greeting) keep to the Fen Causeway
Another more archaic option dispenses with the hyphen question altogether and employs the umlaut convention common in Germanic languages where a vowel following another vowel influences the pronunciation of the second vowel. Thus the words above become coöperation, coöperative, coöp, and coöper. However, this is convention is generally not used today.
In all cases, assume that in a dispute between american and english, the american use is wrong. ;-)))) keep to the Fen Causeway
Recently I was recalling a Spanish verse that goes Qué descansada vida la del que huye del mundanal ruïdo... where ruido s correctly pronounced as rUI-dO but the poet needs three syllables to preserve the meter and so he forces rU-Ï-dO which is signalled in writing by using a dieresis. This is from the 16th century, when the preservation of meter was foremost in poetry. Nowadays, with free verse, people don't bother with meter and so they don't have a need to do violence to words with dieresis. Can the last politician to go out the revolving door please turn the lights off?