HINT: Look at the letters around the edge.

Explanations:

6A Traumatise second vehicle – Second = S, vehicle = CAR => SCAR = traumatise

7A Intertwine two doctors for privacy law – Two doctors = GP and DR, intertwining them => GDPR = privacy law

8A Market the Bill Bryson (in brief) without preparing speech – Market = AD, the Bill Bryson (in brief) = LIB => AD LIB = without preparing speech

9A Forty confused fatherless party – Confused = anagram of FORTY, fatherless = anagram of ORTY => TORY = Party

11A Time I harmonise – Time = T, I = ONE => TONE = harmonise

12A Clasp demi-snake? – Demi = take half of clASP => ASP = type of snake

14A Connected with no students who are tattooed – Connected = LINKED, students = L, no = remove the L => INKED = tattooed

16A Right in wonky bolt – Right = R, wonky = SCEW => SCREW = bolt

18A Go downhill fast in overseas kingdom – In = hidden word in overseaS KIngdom => SKI = go downhill fast

20A Consider Scottish island – Consider = Scottish island => MULL

22A Link unending tea – Link = CHAIN, unending = remove the last letter => CHAI = tea

24A Draw out stateless, nameless actor – Draw out = EXTRACT, stateless = remove CT (Connecticut) => EXTRA = nameless actor

25A Odd Italian Banks? – Odd letters of ItAlIaN => IAIN as in Iain Banks

26A To court after my old man made deal with the devil – Court = CT, my old man = PA => PACT = made deal with the devil

1D Parrot bearing components of carbohydrates – Bearing = E, components of carbohydrates = COH => ECHO = parrot

2D 10 cartilaginous fish find possibly broken bones – 10 = X, cartilaginous fish = RAY => X-RAY = find possibly broken bones

3D First priest’s taxes for the poor makes coins disappear – First (of) Priest’s = P, taxes for the poor = ALMS => PALMS = makes coins disappear

4D Regular Lego batman briefly loves community of his husband – Regular = alternate letters of LeGo BaTmAn, briefly = remove last letter => LGBT = community of his husband

5D Penwomen confused, not in the northwest, by instructions on a parcel – Confused = anagram of PEnwOMEN, not in the northwest = remove NW => OPEN ME = instructions on a Parcel

10D Korean soldier swings without clubs – Swings = ROCKS, clubs = CS => ROK = Korean Soldier

12D Even TARDIS notices… – Even letters of tArDiS => ADS = notices

13D Athenian character is back after first Philippian – IS back = SI, first Philippian = P => PSI = Athenian Character

15D Unreal – lost before network? – Lost = anagram of UNREAL => NEURAL precedes network in common phrase

17D Posh sun god is initially happy – Sun god = RA, initially Happy => RAH = posh

19D Japanese chick at such a hidden place – Hidden place = hidden word in chicK AT SUch => KATSU = Japanese chick

21D Novice moving tree fast – Novice = L, moving tree = ENT => LENT = fast

22D Mafia boss changes guitar tuning – Mafia boss = CAPO = changes guitar tuning

23D Curve leaderless stride – Stride = MARCH, leaderless = remove M => ARCH = Curve