Sadly, I don’t have much to add from the title to this question: does œ exist in English, such as in the word manœuvre?
The same question may also apply to what the French call the “e dans l’a” (e in the a), the æ, in addition to what they call the “e dans l’o” (e in the o), the œ — at least as far as the French part is concerned.
First, be aware that manoeuvre is now normally spelled maneuver in America, and indeed, has fallen behind maneuvre in England. Even the Economist (but not the Œconomist :) uses maneuvre now.
The general answer is that œ is considered a mere typographic ligature in written English, not a lexical ligature as it is in French. See this answer for more about all that.
It is the modern custom to print all instances of œ as oe in English. Indeed, the OED switched its custom from the ligated digraph to the separated form when it went from its 2nd to its 3rd edition.
Therefore, for example, these words are all now typically printed differently. Notice how in some instances, the oe reduces to e. Although increasingly common, that reduction is by no means universal, and does not occur in all words, either.
There are two important exceptions to this.
The first is in terms taken from the French and considered “unassimilated” into English. These are typically set in italic. Since œ is a lexical not a typographic ligature in French, when printing French terms it is imprescindible that the ligature be maintained. For example:
Whether to preserve it in Latin terms like casus fœderis or subpœna is more controversial. Usually, it is not.
The other important exception is when printing phonetic or phonemic transcriptions in the International Phonetic Alphabet (IPA). There are two glyphs that mean something special in IPA, and which must be preserved:
The relevant code points in Unicode are:
Π0152 LATIN CAPITAL LIGATURE OE
œ 0153 LATIN SMALL LIGATURE OE
= ethel (from Old English eðel)
* French, IPA, Old Icelandic, Old English, ...
x (latin small letter ae - 00E6)
x (latin letter small capital oe - 0276)
ɶ 0276 LATIN LETTER SMALL CAPITAL OE
* low front rounded vowel
x (latin small ligature oe - 0153)
Note that these are the only code points whose names include the word LIGATURE but which are generally considered lexical not typographical ones. Unlike the others with LIGATURE in their names, these have no decomposed forms that produce two glyphs, nor are they casewise equivalent to the two-glyph version they are currently spelled with in most English words. This is because you would get the wrong results under certain conditions if you did so.
These are the other Latin ligatures in Unicode; all are considered typographic in most languages, and have special decomposition and casing rules:
IJ 0132 LATIN CAPITAL LIGATURE IJ
# 0049 004A
ij 0133 LATIN SMALL LIGATURE IJ
* Dutch
# 0069 006A
ff FB00 LATIN SMALL LIGATURE FF
# 0066 0066
fi FB01 LATIN SMALL LIGATURE FI
# 0066 0069
fl FB02 LATIN SMALL LIGATURE FL
# 0066 006C
ffi FB03 LATIN SMALL LIGATURE FFI
# 0066 0066 0069
ffl FB04 LATIN SMALL LIGATURE FFL
# 0066 0066 006C
ſt FB05 LATIN SMALL LIGATURE LONG S T
# 017F 0074
st FB06 LATIN SMALL LIGATURE ST
# 0073 0074
Note that all of those are there for legacy round-tripping, and no more such things shall ever be added to Unicode. That is because unlike lexical ligatures, typographic ligatures belong in the font, not the code points.
That is why the typographical ligatures given above get split up under decomposition and caseless matching, but the lexical ligatures do not.
The exception to this is when using the (default, untailored) Unicode Collation Algorithm (UCA) for sorting. Now these code points sort next to the two-glyph version, and count as equivalent at the primary strength (that is, whether they are the same letters or different ones, and without regard to case or diacritics).
Here then is the result of sorting the various words pointed out above using the default UCA:
à contre-cœur, amoebae, amœbæ, apnea, apnoea, apnœa, bœuf, casus fœderis, chef d’œuvre, coelacanth, cœlacanth, cri de cœur, diarrhea, diarrhoea, diarrhœa, edema, epopoeia, epopœia, esophageal, estrus, homeopathic, homoeopathic, homœopathic, hors d’œuvre, maneuver, maneuvre, manoeuvre, manœuvre, melopoeia, melopœïa, Mobius, Möbius, Moebius, Mœbius, mœurs, oedema, œdema, Oedipus, Œdipus, œil-de-bœuf, oenologist, œnologist, oesophageal, œsophageal, œstrous, oestrus, œuf en cocotte, œufs sur le plat, œuvre, Phoebus Apollo, Phœbus Apollo, Phoenician, Phœnician, phoenix, phœnix, subpoena, subpœna, vœu, vœux.
It's not unheard of to see Πin English orthography, and was once much more common than it is now. These days though, it would be so rare as to be only barely acceptable, unless the word was clearly being used as a foreign word (e.g. with italics). (I personally use it privately, but I'll change it in anything being sent to an editor even if I'm not writing to a particular style-guide).
Generally British English spellings change it to oe (manoeuvre, foetus) while American spellings change it to e (maneuvre, fetus), this being one of Noah Webster's reforms. This can't be depended upon: Foederal is almost never found anywhere (any later than the 18th Century or so, anyway), and conversely while subpena is found in American spellings, sub poena is more common there.
Æ is mostly similar, though note that the existance of Æ in a modern word can be from two sources - one is from ligatures the same as with Œ, while the other is a separate vowel letter independent of A and E that was once used in Old English and is still used in Danish, Norwegian, Faroese and Icelandic (In Swedish and German it mutated into Ä). If a word is borrowed from one of those languages, but changed to use the English alphabet, then it would be common to see its spelling become ae rather than e even in American spellings.
One place ligatures remain in modern English is the name Encyclopædia Britannica, since that's the way they spell it, and trademarks and personal and company names are not normally re-spelled between different English orthographies.
The ligatures can certainly be found in English, although their use is becoming less common — probably due to the rise in personal computers and the difficulty in using ligatures with a standard keyboard.
OED gives a recent citation showing the ligature:
1977 Lancet 28 May 1140/1: “In shallow diving an over-forceful Valsalva manœuvre may give rise to neuro-sensory hearing loss, with or without vertigo.”
Their citations after 1977 all have separate letters.
