Both "They have a headache" and "They have headaches" are grammatically correct ways to describe the situation where two people are experiencing headaches simultaneously. The choice between them depends on the emphasis and context of the situation: "They have a headache": This emphasizes the shared experience of a singular headache. It suggests that both individuals are experiencing the same type or intensity of headache. "They have headaches": This emphasizes that each individual is experiencing their own separate headache. It implies that the headaches may vary in type or intensity between the two people. Both options effectively convey the idea that multiple people are experiencing headaches, so the choice between them ultimately depends on the specific nuance you wish to convey.

I think I would say： They both have a headache.

‘They have a headache.’ suggests a figurative meaning of ‘headache’ to mean ‘problem’. (They all have the same problem)

1 They have a headache

If you mean that this is happening now and that it doesn't reoccur, then personally I'd go with Peter's reply. Or a variation like "(INSERT NUMBER) of them have a headache." Or "All of them have a headache." Using this is the best option because it makes it clear that more than one person is having a headache at the same time (whether that's a coincidence or in some way connected isn't clear). I think that the plural version would be confusing because it's used when a person suffers from repeated instances of a headache. For example "He suffers from headaches every morning." And saying "They have a headache." might need to be clarified because of the fact that "they" can be used as singular pronoun.