Across:
1 Sounds like (= homophone indicator) someone going to Rome in the future (someone going to ‘Rome’ = ‘roam’ in the future = TIME TRAVELLER) could be a back-packer? (TIME TRAVELLER),
10 This doesn’t take long (INSTANT), for one (I-) school (National School = -NS-) head teacher (head ‘Teacher’ = -T-) to lose the head (r) on rant (‘rant’ without ‘r’ = -ANT) (INSTANT),
11 One (I-) serial (serial) adaptation (= anagram indicator) (serial = -SRAELI) (ISRAELI) of Living in the Middle East (ISRAELI),
12 Lippy type (EDGE) from golf club (wedge) needing no introduction (w) (‘wedge’ without ‘w’ = EDGE),
13 Good (good/moral = NOBLE) Lord! (NOBLE),
15 Most of those in 30 across (most of ‘times’ = TIME) from some Djibouti metropolis (in some ‘DjibouTI MEtropolis’ = TIME),
17 This is rejected by stylish (‘this’ is rejected by ‘stylish’ = syl = SLY), crafty type (SLY),
19 Policeman (cop) blows top (c) (‘cop’ loses ‘c’ = OP-) with jerks (-TICS) (OPTICS) hanging out of the bar (OPTICS),
21 Cowboy (GAUCHO) endlessly lacking (e) in the social graces (‘gauche’ without ‘e’ = GAUCH-) from Ring (-O) (GAUCHO),
22 Streaming (= anagram indicator) photos (photos = HOT SPO-) for distribution, statistically speaking (distribution in statistics = -T) (HOT SPOT), to somewhere with Wi-Fi (HOT SPOT),
23 Writer (E-D) admits won’t make deadline (-LATE-) (ELATED) for High Spirits (ELATED),
25 Found on foot (TARSUS bones in/on foot) coming from an old city in Turkey (TARSUS),
27 Old stager misses alerts (‘old stager’ misses ‘alerts’ = odg = DOG) from a man’s best friend (DOG),
29 Bengals drop ban (‘Bengals’ drop ‘ban’ = egls = LEGS) in stages (LEGS),
30 Type (TIMES type/font) associated with all sides today (‘time” is part of the four solutions on the sides of today’s crossword = TIMES),
31 No work for bookworm (no ‘work’ for/in ‘bookworm’ = boom = BOOM) to pick-up? (BOOM),
34 Desperately hungry (CRAVING) for first course (first ‘Course’ = C-) flipping one’s lid (-RAVING) (CRAVING),
35 Foreign (= anagram indicator) Legion’s (legions = SLOE GIN) flavoured liqueur (SLOE GIN),
36 Finish your drinks gentlemen? (TIMELY WARNING)? And, as I’m on that subject, don’t drink and drive! (TIMELY WARNING).
Down:
2 Popular (IN-) landmark (-SIGHT) (INSIGHT) for those who have horse sense (INSIGHT),
3 Participants in the Cesarewitch (participants in the ‘ceSAREwitch’ = sare) reviewed (= reversal indicator) (sare = ERAS) in The Times (ERAS),
4 Proportions (RATIOS) in starters of ripe avocados tells its own story (starters of ‘Ripe Avocados Tells Its Own Story’ = RATIOS),
5 Conceitedly (VAINLY) produces a vinyl (a vinyl) reproduction (= anagram indicator) (a vinyl = VAINLY),
6 Old instrument (LYRE) from theatres not in Harley Street (‘theatres’ not in ‘Harley Street’ = rlye = LYRE),
7 Key (music key E-) to playing (= anagram indicator) Gaelic (Gaelic = -LEGIAC) (ELEGIAC) produced by mournful type, as the linesman (poet) sees it (ELEGIAC),
8 When prisoner has to think about, at a stretch (TIME TO REFLECT), that momentary glance in The Mirror (TIME TO REFLECT),
9 This doesn’t leave a minute (TIME-CONSUMING) during dinner break? (during dinner break = TIME-CONSUMING),
14 Bloom (BLOSSOM) is the first to blubber (the first to ‘Blubber’ = B-) at bereavement (-LOSS-) of leading (‘Of’ leading = -O-) Frenchman (Monsieur = -M) (BLOSSOM),
16 Is unable to wait for (ACHES for) handicap (= anagram indicator) chase (chase = ACHES),
18 Fixed (= anagram indicator) a flat (a flat = FATAL) that’s deadly (FATAL),
20 Mess around dropping manures (‘mess around’ dropping ‘manures’ = sod = SOD) in the grass patch (SOD),
21 Figured out (GOT) what Alberto Giacometti concealed (what ‘AlberTO Giacometti’ concealed = tog), backfired (= reversal indicator) (tog = GOT),
24 A (A-) key (music key -F-) hang- (hang) -out (= anagram indicator) (hang = -GANH-) for one (-I) (AFGANHI) having roots in Asia (AFGANHI),
26 Photograph (SHOT-) and piece (-GUN) (SHOTGUN) often aimed at the birds in the country (SHOTGUN),
27 Hotel (-H-) in drab (dingy = DING-Y) (DINGHY) part of The Marina (DINGHY),
28 Japanese hostess (GEISHA) in Newbridge is hardworking (in ‘NewbridGE IS HArdworking’ = GEISHA),
32 Sink one’s teeth into (BITE) both sides of bitter chocolate (both sides of ‘BItter chocolaTE’ = BITE),
33 It might generate some interest (LOAN generates interest for the bank) for longshoreman on the banks (for ‘LOngshoremAN’ on the banks = LOAN).