I don't adhere to it. It is dated, perhaps even archaic. But it is still the preferred spelling in some style guides, and it is still used in some publications.
So “deer” has two e’s but no umlaut because they make the same sound but “coördinate” has the umlaut since the second o is a new sound.
I personally like them and wish they would come back if only as a small push back against the horrific ambiguity that is the English language
That's a somewhat naïve take.
...then again, even some Germans are using the diaeresis for this purpose, such as this guy https://en.m.wikipedia.org/wiki/Bernhard_Ho%C3%ABcker - and you can't blame him, because usually "oe" is a transliteration of "ö", and then his name would mean "hump" (as in camel's hump).
The only place I've ever seen the diaeresis used like that is in the New Yorker.
But that's the only not-foreign not-name word from the list I've seen written with the ¨ in Britain. I have see Chloë and Zoë written with it too.
https://en.wiktionary.org/wiki/Category:English_terms_spelle...