The woman was sitting outside Queenswood restaurant in Battersea Square with two men, one believed to be an undercover police officer, when the arrest operation began.
At 2.20pm (local time), an unmarked police car sped into the square followed by a police van which stopped outside the restaurant.
Three police officers, two male and one female, spoke briefly to the woman. She was then searched by the female officer before stepping into the back of the van.
She remained composed throughout and did not remonstrate with the officers.
One eyewitness, who was eating at the restaurant, said: "She was just sitting there really quietly with two men in ordinary clothes either side of her. She was completely calm and passive. She seemed completely unfazed."
Commanders at the Metropolitan Police's Royalty and Specialist Protection Unit launched an immediate review of the security operation surrounding his studies. But the incident will raise fears of what could have happened had the woman been a terrorist.
One former senior officer last night described the shocking breach as "very worrying", but added: "Schools are not fortresses."
The prince only started his studies last Thursday (local time) when he was photographed being dropped off at the school by his father, the Duke of Cambridge.
The Duchess of Cambridge, who is pregnant with her third child and has been suffering from a severe form of morning sickness, told parents she will meet them at the gates when she, too, is able to do the school run.
The security incident involving the woman began unfolding on Tuesday afternoon (local time).
It is understood that she attempted to gain access to the school by posing as a visitor, possibly with a pre-booked appointment.
But she quickly left the building, which opens directly onto a busy road, after being challenged by a member of staff. It is unclear how far inside she got.
Shortly before 2pm yesterday (local time) the woman was seen again in a small leafy square a short distance from the school gates.
Police were alerted and she was held by plainclothes officers until uniformed colleagues arrived to search and arrest her.
One eyewitness, who was eating at Queenswood, in Battersea Square, said she was alerted by two "very agitated" teachers wearing distinctive name badges standing outside.
She realised that a woman had been made to a sit at table at an adjacent cafe and was surrounded by two men in civilian clothes.
"She seemed very placid, completely unfazed by what was going on," she said. "She was silent and did not exchange anything with police at all. She had light dark hair in a ponytail and had a rucksack-type bag over her shoulder.
"A woman officer arrived and searched her and then she stepped into the back of the police van.
"There were several Thomas's staff wearing name badges who looked very agitated. I saw them being shown some passports or a passport. It made me think the woman could be from overseas."
The woman was being questioned on suspicion of attempted burglary at a south London police station last night. Her mental health will be assessed. One line of inquiry is that she is a 'fixated individual' who has become obsessed with the glamorous Royal couple.
Many security experts believe such people pose a greater threat to the Royal Family than terrorists because they are often unpredictable loners.
Questions were raised over security at Thomas's earlier this month after a woman was able to video herself walking inside it.
Battersea resident Sarah Burnett-Moore, 54, filmed inside the £18,000-a-year school while it was being refurbished over the summer. It appears she was able to enter the premises via the same iron gate and main door after they were left open by contractors.
She said she could have walked in with a home-made bomb and many neighbours of the school are concerned about the security implications of the royal pupil.
Residents said they are concerned that security at the school is not adequate for the high-profile pupil. One mother, aged 39, said: "Incidents like this make us all scared. Prince George shouldn't go here. None of the mums around here want him to go to the school because it's putting us locals and our children under threat.
"The school needs better security and gates. Anyone can walk in. I walk past and can just walk inside whenever I like if I wanted to."
In contrast another woman added: "I didn't feel safe before, but now Prince George is at the school I actually feel a lot safer. It means there is more heightened security than there was before.
"I was mugged just down the road and it's one of those places that is one extreme to the other."
Chris Phillips, the former head of the National Counter Terrorism Security Office, said "It does not surprise me,' he said. 'Schools are public places, they have never been fortresses," he added.
"It's very worrying when you get incidents like this. But don't forget Buckingham Palace has had a number of break-ins and that is pretty much a fortress. Whenever George is there are usually armed police there too."
He added that the duke and duchess must balance leading ordinary lives with their security.
Scotland Yard said: "Police were alerted and officers attended immediately after the issue came to light. We are working with the school to review its security arrangements after the incident."
A Thomas's spokesman said: "We are unable to comment due to the ongoing police investigation into this incident."
