Across:
1 Hotel (-H-) in Burma (Burma = R-UMBA) running (= anagram indicator) (RHUMBA) alternative dance? (RHUMBA / rumba),
4 Composition (RHAPSODY) of working (= anagram indicator) shop yard (shop yard = RHAPSODY),
9 Caught (-C-) copper (-CU-) cutting joint (hip joint = HI-P) (HICCUP) – it’s a minor setback (HICCUP),
10 Miles (Miles = MI-LES) welcomes revolutionary (= reverse indicator) vehicle (car = -RAC-) (MIRACLES) – the handwork of the man upstairs (MIRACLES),
12 Wellington’s (Wellingtons = BOOTS) major employer (ARMY) opens (= position indicator) (ARMY BOOTS) for those in 19 down, protecting the little piggies (ARMY BOOTS protecting the little piggies/feet in ‘forces’),
13 Stay away from (EVADE) First Lady (EV-E) over Bill (-AD-) (EVADE),
14 Overseas crime-fighters (SCOTLAND YARD), exercising the record holders (exercising the record holders/prisoners = YARD) after (= position indicator) 26 across (‘Scotland’ = SCOTLAND) (SCOTLAND YARD),
18 Sauce (DRESSING) to go (GO-) with starters of wings, nuts (starters of ‘Wings Nuts’ = -WN) (DRESSING GOWN) – typically, put it on first thing in the morning (DRESSING GOWN),
21 Odd (= anagram indicator) chore (chore = OCHRE) for shady type (OCHRE),
22 Took a seat (SAT-) from Liberal (-L-) in elite (elite = -EL-ITE) (SATELLITE) celestial body (SATELLITE),
24 Superficial (COSMETIC) funny (= anagram indicator) comic set (comic set = COSMETIC),
25 Cops (POLICE) those in 14 across, for instance (POLICE in ‘Scotland Yard’),
26 Bed (-COT-) is large (-L-) in ocean liner (ocean liner = S-AND) (SCOTLAND) produced in the UK (SCOTLAND),
27 Tease (tease = TEA SE-) out (= anagram indicator) primarily for Turtle (primarily for ‘Turtle’ = -T) (TEA SET) and for the Mad Hatters (TEA SET for the Mad Hatters Tea Party).
Down:
1 Practise (REHEARSE) in hospital (ER), retire (= reverse indicator) (ER = RE-) and take it slowly on the journey to one’s final resting place (-HEARSE) (REHEARSE),
2 Novel (UNCOMMON) produced by a foreign (a = UN-) general (-COMMON) (UNCOMMON),
3 Turns up (= reverse indicator), source of illumination (light bulb = BLU-B) about first race (first ‘Race’ = -R-) (BLURB) is produced as publicity for the bookmaker (BLURB),
5 They give some people a clip around the ears (HAIRSTYLISTS) for cutting staff (HAIRSTYLISTS),
6 Our father (‘Our Father …’ PRAYER) is a runner (RUG) (PRAYER RUG) bringing devotees to their knees (PRAYER RUG),
7 Owner of energy company (OILMAN) producing gas from magnolias (‘ gas’ from ‘magnolias’ = mnolia = OILMAN),
8 Of course (YES -), that lot on stag party (-MEN) (YES-MEN) can never say no (YES-MEN),
11 Local bar (local ‘bar’ = JOINT) provides liquid lunch (SESSION) (JOINT SESSION) for all the members from both houses (JOINT SESSION),
15 What Jack saw The Giant consuming (LARGE MEAL) on the dinner table in front of atlas? (on the dinner table in front of Atlas/Greek giant = LARGE MEAL),
16 Pilots (pilots) flying (= anagram indicator) (pilots = POLIT-S) around Channel Islands (CI) arising (= reverse indicator) (CI = -IC-) (POLITICS) from government business (POLITICS),
17 It’s fundamental (INHERENT) to the inner (the inner) working? (= anagram indicator) (the inner = INHERENT),
19 Bullies (FORCES) might (might/power = FORCE-) go to school (-S) (FORCES),
20 Cry out of hypocrisy (‘cry’ out of ‘hypocrisy’ = hypois = PHYSIO) with one of those having to deal with the strain (PHYSIO),
23 Lots of gents (LOOS-) from Spain (-E) (LOOSE) not fixed to the one place (LOOSE).