My guess is that they inject liquid hydrogen into the hot air, so the cooling is actually done in free "air" rather than on the surface of a traditional metal heat exchanger - that would explain how they cool it so fast and help explain how they prevent frost in the heat exchanger during the cooling.
This reformulates the problem into "how do we prevent the injectors from..."
a) Melting (liquid hydrogen inside would help a lot).
b) Preventing the liquid hydrogen inside the injectors from boiling and rupturing the piping and injectors.
c) Avoid all the standard hydrogen problems (e.g. embrittlement).
In this guess, (a) and (b) would be a delicate engineering balance to keep the injectors cold enough to retain their strength yet warm enough to not ice up. (c) has known solutions.
Can't say anything about the plausibility of the scheme you mention, but the diagram in the BBC article contradicts it. There you can see that the primary cooling medium is liquid helium, in a closed circuit and a liquid nitrogen heat exchanger. Obviously the diagram can be just flawed, either as a disguise or unintentionally.
My bad, I did not read the much more informative BBC article before speculating.
Having a closed circuit nitrogen heat exchanger is interesting because it is a heat exchanger... you still need to get rid of the heat somewhere. For the demo (per the BBC), they evaporated some of the nitrogen to get rid of the heat, but that means you need excess nitrogen to boil off enough to reject the heat. I wonder how they plan to get rid of the heat in a real aircraft.
It also makes me wonder how they can have enough surface area to absorb the heat in the precooler but maintain enough airflow to generate the thrust they would need.
I think you are right, its probably some metamaterial like shark skin, where the surface of the exchanger itself is designed to prevent frost crystallization from occurring .. i.e. the metamaterial is geared towards the properties of water at the stage where it goes from fluidic to crystal .. I forget the term .. so the structure of the surface of the exchanger is formed such as to demote crystallization long enough for the pressure forces to remove the liquid hydrogen ..
Perhaps ultrasound?