He was ready to get his head chopped off without the Lion King even being there to say anything, but sure, they would care about healing him when Mordred was just laughing around and Tristan was saying he was envious of getting hit by the light.
The fact that she would even consider to kill him means she doesn't care if Gawain is there or not, nor would the other knights except Agravain who was a bit worried, but decided to still do as the King said. Mordred is also dying just using Clarent Blood Arthur, stationed outside the city, but she doesn't care, the Lion King doesn't care, Tristan doesn't even acknowledge it or Gawain. Do they even care about the knights getting ****** over due to the supplement thingies they ingest to get stronger and that Agravain uses a bunch of later to defeat Lancelot, pretty much killing himself in the process? I sure as hell don't see them caring.