Across:
1 Formal admissions (INAUGURATIONS) of swearing in (INAUGURATION-) school (-S) (INAUGURATIONS),
10 Rising (= reverse indicator) war of words (row = -WOR-) in Kent (Kent) confusing (= anagram indicator) (Kent = NET-K) (NETWORK) system (NETWORK),
11 One from the Panto (pantomime VILLAIN) has large country house (VILLA-) that’s very trendy (-IN) (VILLAIN),
12 A little bit (ATOM) of what’s served up (= reversal indicator) in Palermo takeaway (in ‘palerMO TAkeaway’ = mota = ATOM),
13 It’s easy to take in (SNACK) Nero, the leader (‘Nero’ the leader = -N-) consumed by Fire (fire/sack = S-ACK) (SNACK),
15 Lament (WEEP) from half of One Direction (half of ‘WEst’ = WE-) on record (Extended Play record = -EP) (WEEP),
17 Produces, scrubs, crops (‘produces’, scrubs/without ‘crops’ = due = DUE) as expected (DUE),
19 Model (‘cast’ = ‘caste’ = CAST-E) on the radio (= homophone indicator) welcoming line (-L-) (CASTLE) in The Palace (CASTLE),
21 Turns up (= reverse indicator) retweet (RT = -T-R) about drugs (-E-) bill (tab = BAT-) (BATTER) for smack (smack/hit = BATTER),
22 Endorse (INITIAL) first (I-) Latin (Latin) novel (= anagram indicator) (Latin = -N-TIAL) about the start of the Interregnum (start of the ‘Interregnum’ = -I-) (INITIAL),
23 Allowed to cool, in a culinary sense (RESTED), in desert (desert) heat? (= anagram indicator) (desert = RESTED),
25 Ask for an opinion (poll = -LLOP) from head barman (Attorney General = AG = GA-) backing (= reverse indicator) (GALLOP) what horse will do on the racecourse (GALLOP),
27 Dressing (DIP) for trifle from field trip (‘trifle’ from ‘field trip’ = idp = DIP),
29 Consumed medication, for instance (TOOK), to (TO-) get all right (-OK) (TOOK),
30 Outfits (GEARS) with great (G-) listeners (-EARS) (GEARS),
31 Amidst the amoeba living (amidst the ‘amoeBA LIving’ = BALI) on Indonesian island (BALI),
34 End product (OUTCOME) is no longer stylish (OUT-) for company (-CO-) and yours truly (-ME) (OUTCOME),
35 Port (P-) drink (cider) knocked back (= reverse indicator) (cider = -REDIC-) with tea (-T) (PREDICT) – hazard a guess what it is? (PREDICT),
36 Little wonder (INFANT PRODIGY) one has a childish gift (PRODIGY) after (= position indicator) new delivery (INFANT) (INFANT PRODIGY).
Down:
2 Sat in on (sat in on) exchange (= anagram indicator) (sat in on = NATIONS) one diplomatically joining the institution that is United? (United NATIONS),
3 Half the food (half the ‘FOod’ = -FO-) in America (US = U-S) (UFOS) produced by crafty types, the subject of conspiracy theorists (UFOS),
4 A French (a = UN-) family (-KIN-) from 6TH Arrondissement (6TH letter of ‘arronDissement = -D) (UNKIND) can be ruthless (UNKIND),
5 Nan leaves in advance (‘nan’ leaves ‘in advance’ = idvace = ADVICE) of guidance (ADVICE),
6 Do nothing (IDLE) with smelt from mild steel (‘smelt’ from ‘mild steel’ = ilde = IDLE),
7 Closest (NEAREST) to eastern (eastern) dish (= anagram indicator) (eastern = NEAREST),
8 Popular (IN-) type from a forecourt (-CAR-) showing off (= anagram indicator) a Citroen (a Citroen = -CERATION) (INCARCERATION) from lock-up (INCARCERATION),
9 It’s in poor taste (INAPPROPRIATE) for Vogue (IN-) to take it without permission (-APPROPRIATE it) (INAPPROPRIATE),
14 American (A-) loses the head (d) with fine powder (‘dust’ without ‘d’ = -UST-) and draught (air) served up (= reverse indicator) (air = -RIA) (AUSTRIA) in Central Europe (AUSTRIA),
16 Article (an = A-N) about fiction (-LIE-) (ALIEN) produced by foreigner (ALIEN),
18 Leaves (SALAD) lover from El Salvador (‘lover’ from ‘El Salvador’ = salad = SALAD),
20 Put a stop to (END) presidents ignoring priests (‘presidents’ ignoring ‘priests’ = den = END),
21 Suitcase (BAG) for characters up (= reverse indicator) for taking a break (characters ‘takinG A Break’ = gab = BAG),
24 Disturb (= anagram indicator) hornets (hornets = SHORTEN) by a bridge? (a bridge = abridge = SHORTEN),
26 Checking out (LEAVING) exam (LEAVING Cert),
27 Rested (rested) awkwardly (= anagram indicator) (rested = DESERT) in ditch (ditch/abandon = DESERT),
28 Strip poker removes kits (‘strip poker’ removes ‘kits’ = rppoer = PROPER) – how is that ladylike? (PROPER),
32 Some plans of action (some ‘planS OF Action’ = SOFA) for those looking for seats in The House (SOFA),
33 Override a decision in part (‘overriDE A Decision’ in part = DEAD) that’s not working (that’s DEAD).