June 9th, 2021

Times 27999 - a gentle end to the 27th millenium.

I would have come close to a PB with this puzzle, until I was slowed down by the 23d and 26a crossers, and 21a which required all the alternate letters in place before I saw what was needed. Two long anagrams can get you off to a fast start. I was a little uncomfortable with the definition at 21a.
EDIT: it seems I was lucky, or just on particularly good form, as most people below found this hard! It's unusual for me to be speedier than most, but even a blind squirrel finds a nut now and then.

1 Fortunate I am on horse: no fun here on bike? (12)
COBBLESTONES - COB (kind of horse) BLEST (fortunate) ONES (I am, one's).
9 Chinese switch positions in capital (5)
HANOI - HAN for Chinese, O and I for off and on positions.
10 Grasping toff has a heart of gold? (5,4)
MEANS WELL - MEAN (grasping) SWELL (toff).
11 Driving competitively, squeezing teammate into roundabout, perhaps (8)
RALLYING - ALLY (teammate) inside RING = roundabout perhaps.
12 Firm resistance: read all about it (6)
STURDY - R for resistance inside STUDY = read.
13 Your extremely evil master’s twisted jesting (8)
DROLLERY - All reversed (twisting): YR (your) EL (ends of EviL) LORD (master).
15 To be ranked worse than F went down badly (6)
GRATED - if you're G RATED you're worse than F.
17 Perhaps designer garments last for one season — gets darker clothes (6)
DENIMS - E N the last letters of one season, inside DIMS = gets darker.
18 Won’t share: had to drink initially enormous amount of alcohol (8)
HOGSHEAD - If you HOG sometning you're not keen to share it; so HOGS = won't share, HAD has E = initially enormous, inserted. A hogshead is a big barrel, its actual volume seems to depend on whether it is for wine, beer, or spirits. 54 gallons for beer, 63 for wine, usually.
20 Memory returning, fool is in confused state (6)
MORASS - ROM (read only memory) reversed, ASS = fool.
21 Frenchman’s duck, caught by a favourite opener (8)
APERIENT - a bit obtuse and my LOI. A PET = a favourite, insert RIEN which means nothing in French. An aperient is a laxative, hence an "opener" of a sort.
24 Old browser I rejected as you are texting way ahead (9)
STEGOSAUR - ST (way, ahead meaning first part of the answer); EGO (I), SA (rejected as) UR (you are in text language).
25 Having got bigger reaction to bad joke perhaps when broadcast (5)
GROWN - sounds like GROAN = reaction to bad joke.
26 Going through endlessly harsh part of mountain, note it took its toll (6,6)
SEVERN BRIDGE - SEVER(E) = endlessly harsh, RIDGE = part of mountain, N.B. = note is inserted ("going through"). I had the bridge bit a while before seeing which bridge we were talking about. The first Severn Bridge was a toll bridge from its opening in 1966 to 2018 when it became "free to enter Wales".
1 Girl in school stuck with another (7)
COHERED - HER (girl) inside CO-ED (school).
2 Having burned oneself very badly, an anxious case (6,2,6)
3 Neglect to send out Times: they haven’t taken orders (5)
LAITY - LAXITY = neglect, remove the X = times.
4 House not ever needed as college (8)
SEMINARY - SEMI (semi-detached house) NARY meaning never a one.
5 Female avoiding wife’s state (4)
OMAN - WOMAN loses W for wife.
6 Do they spread the cost of a superficial education? (4,5)
EASY TERMS - witty cryptic definition.
7 Shy contender, he turned out founder of a dynasty (5,3,6)
HENRY THE SECOND - (SHY CONTENDER HE)*. First king of the House of Plantagenet, 1154 to 1189.
8 You being laid in earth was excessively sickening (6)
CLOYED - YE = you inside CLOD = earth.
14 Rock and roll single gripping me (9)
LIMESTONE - LIST (roll) ONE (single) has ME inserted.
16 Composer’s astonishing feat, in captivating monarch (8)
COUPERIN - COUP =astonishing feat, ER = monarch, IN. French baroque composer, François Couperin.
17 Prosecutor not to reveal being a silk (6)
DAMASK - DA (district attorney) MASK (not to reveal).
19 Period of relaxation having taking a knock largely with alien invasion (7)
DETENTE - DENTE(D) = taken a knock largely, insert ET the usual 'alien'.
22 I hear you pulled up some more gorse (5)
ROGER - hidden reversed as above.
23 Worry as distinct period of change announced (4)
FAZE - sounds like "phase" = distinct period of change. Collins gives FAZE: to disconcert, worry, disturb. I suspect many people use the word in this sense meaning worry thinking it is spelt 'phase'.