Threat wounds are placed one at a time. If, say, you are assigned 3 threat damage and Theoden is exhausted, you could assign the first threat wound to Theoden, killing him, which then triggers his text. So you would play the new companion from that new trigger, then you resume from where you left off. The remaining 2 threat wounds could indeed be assigned to the new companion.
Edit: It seems I misread your initial question. If Theoden is the one getting killed and the threat wounds are being placed due to his death, all of the threat wounds are assigned before Theoden can play the new companion.
Both Toldea and placing the wound from losing the skirmish are required actions, so the FP chooses which one will occur first. If FP chooses for the losing wound to occur first, then Theoden would get killed and play his new companion like normal, then the second required action (Toldea's text) kicks in and the new companion could be exerted if FP chooses.