Travel Agent booked flight in error at my cost

Flitton

British Airways Cancelled direct flight Buenos Aires to Heathrow via my travel agent. Travel agent relayed BA offered a flight from Rio to Heathrow only, but not From Buenos Aires to Rio. (Buenos Aires border closed at this point no direct flights)
Agent booked a flight at MY COST Buenos Aires to Rio to be able to connect to the BA flight, I  tried to claim a refund from BA for a flight BA to Rio that was not offered, BA refused saying that they did offer such a flight to the agent.
Who do I seek compensation from is it the agent or BA?
The agent is adamant they are not at fault as is BA (although BA has shown me evidence that such a flight was booked by them)

Life__Goes__On

Did you get any money back from the cancelled flight (BA-Lon)?  or was it a swap for the Rio-Lon)?

Flitton

Hi, no money back as British Airways are saying they honoured the cancellation Buenos Aires to Rio to  Heathrow with the travel agent, the dispute is Travel Agent are saying they did not, so they booked me a flight to Rio from Buenos Aires at a cost to me £625+ this is the compensation I am looking for.

Life__Goes__On

Was it just a one way ticket  or part of a return? How much noticed did BA give on cancelled flight?
EDIT
I don't think those question really matter BA  should have rerouted you,  so you will be able to claim that extra flight from BA.

Flitton

When I made a claim against BA  they where adamant they did re route me, they did provide evidence of this, which I provided to the travel agent
The day the booking was made by the travel agent there was obvious miscommunication between BA/Lime management & the travel agent. 
The travel agent say BA did not book the flight hence why the travel agent booked another flight.

Life__Goes__On

What method did you use to pay travel agent for extra ticket?

Flitton

Bank card not a credit card unfortunatly, the money was paid to an  Argentinian airline.
The travel agent booked it offcourse  

Life__Goes__On

Flitton said:

Bank card not a credit card unfortunatly, the money was paid to an  Argentinian airline.
The travel agent booked it offcourse  

Where is the travel agent based?  Is the transaction billed as the airline or travel agent?

What evidence did BA provide to show they rerouted you?

Flitton

Travel Agent is in the UK, transaction is showing as the Argentinian Airline, I now realise why they would not invoice me to pay them direct. 
BA provided a data sheet of all my bookings also on my BA account 24 hours prior I received notification to check in, I highlighted this to the travel agent at the time, they told me to ignore it, as they had booked the flight now, and it was an error on BA 
 I was stranded in Buenos Aires, locked in a room, stressed just wanting to get home, borders where closing fast. I was relying on information from the travel agent who booked my original flights

Life__Goes__On

Did the travel agent tell you they were buying the extra ticket before they bought it?

Flitton

Yes they did, I had to take  their word at the time, I have an email saying that BA where no being helpful and they needed to get me home as soon as, therefore they needed to book the flight, they then rung me for my Bank card details.