Does anyone know the answer to the following denied boarding question?

Our original booking was with AirEuropa from EZE (Buenos Aires, Argentina) to LGW (London Gatwick, UK) via MAD (Madrid, Spain).
The first leg UX42 (EZE to MAD) was delayed by 5 hrs and therefore AirEuropa rebooked our ticket as we were travelling with 2 small kids.
The following flights were booked by AirEuropa and a FIM (flight interruption manifest) issued:

Leg 1 IB6842 EZE to MAD
Leg 2 IB7440 MAD to LHR Operated by BA BA0455

The boarding passes for both flights were printed and issued by Iberia in EZE airport.
We boarded our first flight in EZE and arrived 5:30am in MAD and made our way to the gate for our second leg.
5 minutes before boarding we were told (in a fairly aggressive tone) by Iberia ground staff that BA didn’t allow us to board, because our booking/reservation was invalid as no agreement exists between AirEuropa and BA.
Obviously we were stunned and confused, which only got worse by the ground staff not being able to answer some basic questions.

Q: We have boarding passes printed by Iberia in Argentina
A: The boarding passes have been issued by mistake as the booking by AirEuropa is invalid. (the reservation shows the IB7440 flight number instead of the BA455 number)

Q: it’s a codeshare flight between Iberia and BA, and our reservation is with Iberia and not BA according to the reservation document (FIM).
A: Iberia and BA only work together at a marketing level, the actual flights are operated by different airlines and reservations are made with the individual airlines
The boarding passes have both flight numbers (BA0455 and IB7440) printed and the ground staff was wearing Iberia uniform.

She walked away after 2 minutes and we had no other option than to continue our conversation at the Iberia customer service desk.
We were told to go to the AirEuropa customer service to request a new booking on another Iberia flight and collect our luggage as we had to recheck our luggage on the next flight.
It was a logistical nightmare having to collect the luggage in T4 and having to travel to T2 (a bus ride away) to visit the AirEuropa customer service desk to request a new reservation. All of this with a baby and toddler in tow.
The AirEuropa customer service desk was very confused with the whole situation and wondered why on earth neither BA or Iberia contacted AirEuropa to sort out any problems ahead of our departure from Madrid, however they immediately made a new reservation with Iberia for the first available flight to avoid further delays which unfortunately was not until 13:15 (a delay of nearly 6 hours).

Our attempt to check in with our new reservation (MAD-LHR) failed miserably due to our ‘live’ reservation on the ‘code-shared’ BA455/IB7440 flight.
In the end, it took Iberia 30-45 minutes to cancel the original reservation for flight BA455/IB7440 and a new reservation was made.

Apologies for the long essay, however I wanted to provide as much detail as possible.
The burning question is, which airline is responsible for the denied boarding without the passengers being at fault and liable for the compensation to be paid according to European Regulations.