On 24/07/2023, Assistant Commissioner Matt Twist, a senior officer in the Metropolitan Police Service (the "Met"), released a statement giving the police's account of the incident and the "legal basis for her arrest" as requested by the OP.
It is quite lengthy but I have resisted editing or redacting it to ensure the whole statement is available to those users (like me) that would rather not follow anonymous links. I have also resisted emboldening any of the text to allow users to make up their own minds without any unconscious bias on my part.
Finally, I have left my original answer as is for posterity.
The statement in full reads as follows (with the description of what caused the arrest bolded):
“It is clear from the video that has been shared online that this incident was distressing for the woman involved and particularly for her child. We understand why it has prompted significant public concern and we want to be transparent about our position and the role of our officers.
“Officers from the Met’s Roads and Transport Policing Command were supporting TfL ticket inspectors on a pre-planned operation in Whitehorse Road, Croydon on Friday, 21 July. As buses pulled into the stop, TfL inspectors would check the tickets of those onboard and also those getting off.
“Anyone without a valid ticket is required to provide their details to a TfL inspector so a penalty fare can be issued. This is not a policing matter. Officers only become involved where details are not provided or where someone tries to leave when challenged.
“The woman involved in this incident was asked to provide her ticket as she got off the bus, but did not do so. She was spoken to by a TfL inspector, then by a PCSO and finally by a police officer. She continued to try to walk away and did not provide her ticket for inspection.
“She was arrested on suspicion of fare evasion and was handcuffed. When officers were able to take her ticket from her so that the TfL inspectors could check it, they were able to confirm it was valid. She was immediately de-arrested and her handcuffs were removed.
“Throughout the incident, the child was comforted by a PCSO who immediately recognised his distress. Anyone seeing how upset he was would be moved by this, and we regret any impact it may have on him.
“We recognise that the use of handcuffs can be a cause of concern, particularly given the context of this incident and the type of offence involved, but when a person is trying to physically leave an incident it is an option officers can consider. All uses of force must be proportionate and necessary in the circumstances.
“Ticket inspection operations of this nature are difficult. They place police officers in direct confrontation with frustrated members of the public and could escalate what would otherwise be civil matters to a different level.
“This incident raises questions about the extent to which officers are having to intervene in this way when supporting TfL in their operations. We will now work with TfL to ensure that the balance is right between officers tackling the most serious crime on the transport network and supporting their own operations to ensure revenue protection.
“An initial review of the officers’ actions did not identify any conduct matters but we will reflect on it carefully, in discussion with communities locally, to urgently identify any opportunities to do things differently.
“Given the level of community concern generated we believe it is in the public interest to voluntarily refer the matter to the Independent Office for Police Conduct to review.”
A breach of the Regulations is a summary offence. As such, a suspect may be arrested if all the relevant conditions at section 24 Police and Criminal Evidence Act 1984 are met.
See section 67 Public Passenger Vehicles Act 1981:
Penalty for breach of regulations.
Subject to section 68(1) of this Act [i.e. reasonable excuse], if a person acts in contravention of, or fails to comply with, any regulations made by the Secretary of State under this Act and contravention thereof, or failure to comply therewith, is not made an offence under any other provision of this Act, he shall for each offence be liable on summary conviction to a fine not exceeding level 2 on the standard scale.
And see Paragraph 7 Public Service Vehicles
(Conduct of Drivers, Inspectors, Conductors and Passengers) Regulations 1990:
(2) ... every passenger on a vehicle being used for the carriage of passengers at separate fares shall–
(a)declare, if so requested by the driver, inspector or conductor, the journey which he intends to take, is taking or has taken in the vehicle;
(b)where the vehicle is being operated by the driver without a conductor–
(i)save as provided in (ii) below, immediately on boarding the vehicle, pay the fare for the journey he intends to take to the driver or, where appropriate, by inserting in any fare-collection equipment provided on the vehicle the money or token required to pay that fare; or
(ii)if otherwise directed by the driver, an inspector or a notice displayed on the vehicle, shall pay the fare for his journey in accordance with the direction;
(c)where the vehicle is being operated by the driver with a conductor, pay the fare for the journey which he intends to take, is taking, or has taken in the vehicle to the conductor immediately on being requested to do so by the conductor or an inspector;
(d)accept and retain for the rest of his journey any ticket which is provided on payment of a fare in accordance with sub-paragraph (b) or (c);
(e)produce during his journey any ticket which has been issued to him either under sub-paragraph (d) or before he started his journey for inspection by the driver, inspector or conductor on being requested to do so by the driver, inspector or conductor; and
(f)as soon as he has completed the journey for which he has a ticket, either–
(i)leave the vehicle; or
(ii)pay the fare for any further journey which he intends to take on the vehicle.
Paragraph 7 covers numerous alternative scenarios and as I have not seen the video, nor know the surrounding circumstances, I have emboldened the more likely alleged breach(es). I will review this if/when more details become available.