I don't know if this is your particular problem, but it's widely accepted that drinking tea on an empty stomach will make you nauseous. The fact that the addition of milk alleviates these symptoms makes me feel this is perhaps your problem. Milk contains fats, proteins, and calories among other things - so not only tea is entering the stomach. The milk may be providing an alternative mechanism for you to safely digest the tea with nutrients present. Try eating a bagel or something before your next cup of black tea - like right before, not an hour before.
Excedrin Migraine also does it to me
Interesting auggy, especially seeing how Excedrin Migraine contains 65 mg of caffeine in each pill - or maybe that's why you mentioned it in the first place. The caffeine is supposedly present to increase the pharmicokinetics of the pain reliever, however it's my personal opinion that it's there for people experiencing caffeine withdrawal headaches - something too few people are aware they are susceptible to. I could see how your caffeine sensitivity interacts the way it does with this medication.