Across:
1 Play (= anagram indicator) a cello (a cello = LOCALE) in familiar setting (LOCALE),
4 Old jaguars (USED CARS) from uninhabitable sprawling summits (‘Uninhabitable Sprawling’ summits/first letters = US-) with wild (= anagram indicator) cedars (cedars = -ED CARS) (USED CARS),
9 Seth leaves theorising (‘Seth’ leaves ‘theorising’ = oriing = ORIGIN) to Dawn? (dawn/birth = ORIGIN),
10 That’s embarrassing (ABASHING) for (A-) party (-BASH-) in (-IN-) Government (-G) (ABASHING),
12 Anonymously (INCOGNITO) and illicitly copy (-COG-) popular (in) review (= reversal indicator) (in = -NI-) dividing teachers (Irish National Teachers’ Organisation = INTO = IN-TO) (INCOGNITO),
13 Tailor-made without model (‘tailor-made’ without ‘model’ = taira = RAITA) for a dippy Indian type (RAITA).
14 Move (SWIZZLE) staff (staff/rod = STICK) (SWIZZLE STICK) for stirring up the old-fashioned types? (SWIZZLE STICK for stirring up an “Old Fashioned” cocktail),
18 Posing (SITTING) with The Donald’s (Donald Duck / DUCKS) (SITTING DUCKS) patsies who are bound to be exposed (SITTING DUCKS),
21 Staring into space (musing), scratching the head (m) (‘musing’ without ‘m’ = USING) – that’s not clean and it’s illegal (USING drugs),
22 Turns up (= reverse indicator), local fool (nit = T-IN-) welcomes incense (-ANGER-) from Oriental (East = -E) (TANGERINE) citrus fruit (TANGERINE),
24 Argue over trifles (PETTIFOG) as darling (PET-) knocks it (it = -TI-) back (= reverse indicator) with London’s famous pea soup? (“pea soup” fog = -FOG) (PETTIFOG),
25 Loud noise (HUBBUB) from centre of activity (hub = H-UB) around university (-U-) building offering accommodation (bed and breakfast = B&B = -BB-) (HUBBUB),
26 Make a clean sweep (SANITISE) to debug the premises (SANITISE),
27 It’s tough (SEVERE) to break it off (SEVER-) with sweetheart (‘swEet’ heart = -E) (SEVERE).
Down:
1 So polite (so polite), flexible (= anagram indicator) (so polite = LOOPIEST) or most eccentric? (LOOPIEST),
2 Greek letter (CHI-) is not the first (‘Not’ the first = -N-) Greek letter (CHI-) to new (-N) (CHIN-CHIN) and to absent friends (to absent friends = a toast = CHIN-CHIN),
3 It’s stretching the truth to say (LYING) Bob has no part in lobbying (‘Bob’ has no part in ‘lobbying’ = lying = LYING)
5 Very large area (SUBCONTINENT) in U-boat (SUB-) with elements (content = -CONT-ENT) found around at home (at home = -IN-) (SUBCONTINENT),
6 Lost power (died = DI-ED) over endless (end ‘lesS’ = -S-) capsized (= reverse indicator) ship’s goods (cargo) briefly (o) (‘cargo’ without ‘o’ = carg = -GRAC-) (DISGRACED) dragged through the mire (DISGRACED),
7 Is bitter (ACIDIC) with accountant (CA) over (= reverse indicator) (CA = AC-) the papers (-ID- papers) getting one (-I-) caught (-C) (ACIDIC),
8 Wink (SIGNAL) from Bill (SIGN-) associated with American (-A-) First Ladies (first ‘Ladies’ = -L) (SIGNAL),
11 Those who employ Poles (pole in FIRE STATIONS) providing work for those who can go to blazes (FIRE STATIONS),
15 Mood over time (ZEITGEIST) affects (= anagram indicator) size? Get it! (size get it = ZEITGEIST),
16 Lawyer (-SCRIB-E -) consumed by old writer (SCRIB-E) (SCRIBBLE) producing writing that’s difficult to read (SCRIBBLE),
17 Labs seem (labs seem) unusual (= anagram indicator) (labs seem = ASSEMBLE) to construct (ASSEMBLE),
19 Usurp (usurp = RU-PUS) criminal (= anagram indicator) concealing Mafia leader (‘Mafia’ leader = -M-) (RUMPUS) in Donnybrook (donnybrook/uproar = RUMPUS),
20 Small foxy type (KITTEN) of gear (KIT-) is half rotten (half ‘rotTEN’ = -TEN) (KITTEN),
23 Duck (ELUDE) in the swampy water (mud) heading off (m) (‘mud’ without ‘m’ = -UD-) in flowing (= anagram indicator) Lee (Lee = EL-E) (ELUDE).