>> I'd think most dialects of English have /ö/
>That seems unlikely, if you mean present-day dialects.
It is found in NZ-speak There is a description of the name 'Turner' being
pronounced almost exactly as the German word Töne. NZ-speak has been called
the only known language in which there is a single front rounded vowel,
which is not /y/. (I've lost the source of that, but it makes sense. /u/
is a mid-u, neither truly back nor front.)
Peter