My workings follow, keyed by clue, definition SOLUTION
1 Leaving first university, melancholy man beaming (9)
REFULGENT A melancholy man is a RUEFUL GENT. Strike the first U(niversity)
9 Charge soldier to retreat into Scottish island (7)
ARRAIGN If a soldier retreats, he becomes an IG, which leaves you with a Scottish island which must be ARRAN
10 With pressure, burden tradesman (7)
PLUMBER Burden translates (as a verb) to LUMBER, tagged onto P(ressure)
11 Pub welcomes Round Table, at last united (2,3)
IN ONE I think the definition’s OK. The wordplay: pub: INN, round: O, and tablE at last, assembled as instructed, with the rounded and table treated as separate items
12 Dog looking ashamed, having fouled street (3,6)
RED SETTER I’m not sure how a dog gets to be RED faced, but then it doesn’t have to. SETTER comes from a mixed (“fouled”) version of STREET
13 Parts of body that may be found in field (3,4)
LEG SLIP The body parts we need are LEGS and LIP. I hope you didn’t waste too much time trying to work out where the S came from. Leg slip in cricket is a catching position behind batsman and stumps on the side towards which the batsman points his bottom.
15 Electrical inventor not needing most of site’s laboratory (5)
TESLA The great man is found hiding away in siTE’S LAboratory.
17 Rude order to servant to leave fruit (5)
MANGO A peremptory order to a servant might be MAN, GO, though I can think of worse.
18 Doubtful apostle could be so familiar as a soldier? (5)
TOMMY The doubting apostle would be Thomas (John 20.21), and his familiar version gives Tommy (Atkins), the archetypical British soldier
19 Annoy English deputy in US made to return (5)
PEEVE English provides an E, the American deputy id the VEEP (Vice President). Together and “made to return” they provide our answer,
20 Tower with record inscribed in column (7)
STEEPLE A column might well be a STELE, and a record is an E(extended) P(lay)
23 Make fuss and prepare to strike blow on ear (5,4)
RAISE CAIN In order to inflict a chastising blow you would first need to raise (your) cane, which our answer sounds like
25 Such lite food served here? (5)
LOCAL Or lo-cal(orie). Both lite and lo–cal are informal spellings
27 If wearing small skirt, it will ride up in the cold (3,4)
SKI LIFT IF is surrounded by S(mall) KILT
28 Writhed as tragedy unfolded (7)
GYRATED an anagram (“unfolded”) of TRAGEDY
29 Thoroughly European, envy rich criminal (5,4)
EVERY INCH E(uropean) plus an anagram of ENVY RICH, indicated by “criminal”
1 Bear forward: game on! (6)
RUPERT Forward as in cheeky gives PERT, put the game for thugs played by gentlemen, R(ugby) U(nion) on top. Rupert Bear (never with a the) first appeared in the Daily Express on 8th November 1920, where with pals Bill Badger and Edward Trunk he attempted to solve the Case of the Princess in the Tunnel.
2 In illness, one poor student swallowing at first only a little liquid (5,5)
FLUID OUNCE The illness is FLU, one is I, poor student DUNCE and then the first letter of Only remains to be inserted.
3 Looking up pet rodent, avoiding first dry old textbook (8)
LIBRETTO. Otters make poor pets, but GERBILS are OK, and they’re the ones you want.. Reverse as indicated by looking up, and drop the first letter. Dry is T(ee) T(otaller), and O(ld) completes.
4 Say, soak to soften wader (5)
EGRET Say is EG, and RET is to soak in order to soften
5 More peaceful path for medieval conqueror (9)
TAMERLANE 9 April 1336— 18 February 1405 More peaceful is TAMER, and path LANE. Like Christopher Marlowe, I wanted to put a B and an I into his name. But this was a case of trust the cryptic.
6 Fish commonly taking wrong direction (6)
ERRING Not exactly a homophone in any known dialect, so a written H-less version of herring.
7 Vehicle is large, in my opinion (4)
LIMO L(arge) I(n) M(y) O(pinion)
8 Use staff meanly? Make little of losing pounds (8)
UNDERPAY Make little of: UNDERPLAY with the £ missing.
14 Check last piece of editorial copy (10)
LIMITATION The last piece/letter of editoriaL plus IMITATION instead of copy
16 Replace leather worn by individual: not on! (9)
SUPERSEDE SUEDE for leather is “worn” by PERSON (individual without the ON.
17 Slander about one’s not put straight (8)
MISALIGN Slander (verb) provides MALIGN, surround I’S (one’s) with it
18 Impetuousness may finally wear me out — time up! (8)
TEMERITY May finally is Y. Add TIRE ME for wear me out and T for time, all of the “up”, or reversed (it’s a down clue)
21 Dish containing a relish (6)
PALATE Dish is PLATE, and A is contained. I couldn’t, but Chambers and Shakespeare make the direct comparison of palate and relish.
22 Grab and beat up in school (6)
SNATCH TAN is beat, and up indicates a reversal. School cheerfully provides the SCH setting.
24 Problem children (5)
ISSUE a neat double definition, probably with previous convictions.
26 Run into youngster, 14 (4)
CURB an easy R(un) placed in CUB