They dosed her with forkroot at the docks, a strong enough dose to knock her out (she knows they gave her something, just not what).

KOD Prologue
I think you gave her too much of the forkroot,” the woman went on.

A chill flashed through her. So that was what she had been fed!

Felaana tells Katarine she gave her too much and later Silviana tells her they can't risk giving her doses strong enough to knock her out for the entire night because it will make her sick, instead they give her tiny doses once an hour to keep her power to a trickle.

The effects of her first dose are just beginning to wear off when she gets to the Tower, enough to where she could embrace the Source, but it's been less than an hour since she was dosed therefore there is no way she could Channel with much strength. The women in the coach have no idea what Elaida's plans are so they shield when she wakes up and assume that she'll be sent to the open cells (note that even Leane is given forkroot and shielded while there).

Look over the Verin section again... here's some thoughts to consider:

The timeline begins here: The morning passed slowly. Eventually, iron keys clanked as one turned in the ancient lock

After this Katarine walked in announced she was MoN and sent Egwene with Barasine and the other Red ... along the way Egwene took the forkroot

so starting late as mid-morning or early afternoon we have:

1. Egwene takes a dose walks into her room
2. Verin is there sipping her tea
3. Verin lives for about 1 hour
4. Egwene thinks this while studying the books "The next few hours were among the most disturbing and discomforting in her life." ... "She closed the fat red book, thoughtful, her candle burned nearly down to the base. It was growing late in the day."
5. Nicola enters with soup for Verin
6. Meidani enters and spends some time talking before she demonstrates one of the 100 Weaves and then takes Verin's corps via gateway
7. Before Meidani leaves Egwene glanced at her flickering candle; it was burned nearly all the way down to the table itself ... so what another 30 minutes... clearly it's evening by now.
8. Egwene then goes to meet Siuan in TAR
9. She is awakened by Nicola due to the Seanchan attack And she couldn't channel enough Power to light a candle, let alone fight back.

The timeline basically ends here:

Siuan awoke with a start. Something was wrong. Something was very, very wrong. She scrambled off of her pallet. As she did, a dark figure moved suddenly on the other side of the tent, metal rasping against metal. Siuan froze, embracing the Source reflexively and summoning a globe of light.

Gareth Bryne stood alert, heron-marked steel drawn and ready. He wore only his smallclothes, and she had to keep herself from staring at his muscled body, which was in far better shape than that of most men half his age. "What is it?" he asked tensely.

"Light!" Siuan said. "You sleep with your sword?"

Note that Bryne was also sleeping so it wasn't exactly a random mid-morning romp to TAR for info.

It's night by the time the Seanchan attack ... we're talking about a minimum of 6-8 hours of time after Egwene took the forkroot and still could not Channel! It's obvious that in the first scene it's been a much shorter timeline and the dose of forkroot was actually stronger since it knocked her out!