Well, the 'Daily Mail' is probably the worst paper for scare-mongering and pseudo-science in the UK (every day warns that something causes cancer, whilst something natural cures it, and generally just tries to blame everyone's ails on some trendy scapegoat subject like wind farms, "chemicals" or political correctness).
Also, psychologists have a way of coming up with theories and studies that can't be replicated, as such, more than any other science, its findings and theories should be taken with a huge pinch of salt
With that said, I do believe flickering lights can cause headaches - lots of things can cause headaches, especially lighting. (examples include, diet, posture, gait, asking someone if they have a headache - a good proportion will have a headache when you ask them).
As to whether its all LED bulbs (unlikely), how noticeable the flicker is, how many will be negatively affected, its anyone's guess. Whilst I'm sure there will be some LED bulbs sold that will have noticeable flicker - it seems to be the source (the daily fail) is implying its ALL LED bulbs.